 8.1: classify each triangle problem as AAS,ASA, SAS, SSA, or SSS on the ...
 8.2: classify each triangle problem as AAS,ASA, SAS, SSA, or SSS on the ...
 8.3: classify each triangle problem as AAS,ASA, SAS, SSA, or SSS on the ...
 8.4: classify each triangle problem as AAS,ASA, SAS, SSA, or SSS on the ...
 8.5: classify each triangle problem as AAS,ASA, SAS, SSA, or SSS on the ...
 8.6: classify each triangle problem as AAS,ASA, SAS, SSA, or SSS on the ...
 8.7: solve the given triangles.
 8.8: solve the given triangles.
 8.9: solve the given triangles.
 8.10: solve the given triangles.
 8.11: solve the given triangles.
 8.12: solve the given triangles.
 8.13: solve the given triangles.
 8.14: solve the given triangles.
 8.15: solve the given triangles.
 8.16: solve the given triangles.
 8.17: two sides and an angle are given. Determine whether a triangle (or ...
 8.18: two sides and an angle are given. Determine whether a triangle (or ...
 8.19: two sides and an angle are given. Determine whether a triangle (or ...
 8.20: two sides and an angle are given. Determine whether a triangle (or ...
 8.21: two sides and an angle are given. Determine whether a triangle (or ...
 8.22: two sides and an angle are given. Determine whether a triangle (or ...
 8.23: two sides and an angle are given. Determine whether a triangle (or ...
 8.24: two sides and an angle are given. Determine whether a triangle (or ...
 8.25: two sides and an angle are given. Determine whether a triangle (or ...
 8.26: two sides and an angle are given. Determine whether a triangle (or ...
 8.27: two sides and an angle are given. Determine whether a triangle (or ...
 8.28: two sides and an angle are given. Determine whether a triangle (or ...
 8.29: two sides and an angle are given. Determine whether a triangle (or ...
 8.30: two sides and an angle are given. Determine whether a triangle (or ...
 8.31: two sides and an angle are given. Determine whether a triangle (or ...
 8.32: two sides and an angle are given. Determine whether a triangle (or ...
 8.33: two sides and an angle are given. Determine whether a triangle (or ...
 8.34: two sides and an angle are given. Determine whether a triangle (or ...
 8.35: refer to the following:On the launch pad at Kennedy Space Center, t...
 8.36: refer to the following:On the launch pad at Kennedy Space Center, t...
 8.37: HotAir Balloon. A hotair balloon is sighted at the same time by t...
 8.38: HotAir Balloon. A hotair balloon is sighted at the same time by t...
 8.39: Rocket Tracking. A tracking station has two telescopes that are 1 m...
 8.40: Rocket Tracking. Given the data in Exercise 39, how far is the rock...
 8.41: Distance Across River. An engineer wants to construct a bridge acro...
 8.42: Distance Across River. Given the data in Exercise 41, how far is it...
 8.43: Lifeguard Posts. Two lifeguard chairs, labeled P and Q, are located...
 8.44: Rock Climbing. A rock climbing enthusiast is creating a climbing ro...
 8.45: Tennis. Two friends playing tennis are both 70 inches tall. After a...
 8.46: Tennis. Shocked by the move Player I made in Exercise 45, Player II...
 8.47: Archery. (Recall Exercise 73 in Section 6.1 for context.) An expert...
 8.48: Archery. An archer res two arrows simultaneously toward the target....
 8.49: Bowling. The 68 split is common in bowling. To make this split, a ...
 8.50: Bowling. A bowler is said to get a strike on the Brooklyn side of t...
 8.51: refer to the following: To quantify the torque (rotational force) o...
 8.52: refer to the following: To quantify the torque (rotational force) o...
 8.53: explain the mistake that is made. Solve the triangle Solution: Use ...
 8.54: explain the mistake that is made. Solve the triangle Solution: Use ...
 8.55: determine whether each statement is true or false. The Law of Sines...
 8.56: determine whether each statement is true or false.If you are given ...
 8.57: determine whether each statement is true or false.An acute triangle...
 8.58: determine whether each statement is true or false.An obtuse triangl...
 8.59: determine whether each statement is true or false.If you are given ...
 8.60: determine whether each statement is true or false.If is obtuse and ...
 8.61: Mollweides Identity. For any triangle, the following identity is tr...
 8.62: The Law of Tangents. Use the Law of Sines and trigonometric identit...
 8.63: Use the Law of Sines to prove that all angles in an equilateral tri...
 8.64: Suppose that you have a triangle with side lengths a, b, and c, and...
 8.65: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.66: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.67: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.68: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.69: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.70: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.1: the angle sum identity, 180, will be used in solving the triangle. ...
 8.2: the angle sum identity, 180, will be used in solving the triangle. ...
 8.3: the angle sum identity, 180, will be used in solving the triangle. ...
 8.4: the angle sum identity, 180, will be used in solving the triangle. ...
 8.5: the angle sum identity, 180, will be used in solving the triangle. ...
 8.6: the angle sum identity, 180, will be used in solving the triangle. ...
 8.7: the angle sum identity, 180, will be used in solving the triangle. ...
 8.8: the angle sum identity, 180, will be used in solving the triangle. ...
 8.9: solve each triangle.
 8.10: solve each triangle.
 8.11: solve each triangle.
 8.12: solve each triangle.
 8.13: solve each triangle.
 8.14: solve each triangle.
 8.15: solve each triangle.
 8.16: solve each triangle.
 8.17: solve each triangle.
 8.18: solve each triangle.
 8.19: solve each triangle.
 8.20: solve each triangle.
 8.21: solve each triangle.
 8.22: solve each triangle.
 8.23: solve each triangle.
 8.24: solve each triangle.
 8.25: solve each triangle.
 8.26: solve each triangle.
 8.27: solve each triangle.
 8.28: solve each triangle.
 8.29: solve each triangle.
 8.30: solve each triangle.
 8.31: solve each triangle.
 8.32: solve each triangle.
 8.33: solve each triangle.
 8.34: solve each triangle.
 8.35: solve each triangle.
 8.36: solve each triangle.
 8.37: solve each triangle.
 8.38: solve each triangle.
 8.39: solve each triangle.
 8.40: solve each triangle.
 8.41: solve each triangle.
 8.42: solve each triangle.
 8.43: Aviation.A plane ew due north at 500 mph for 3 hours. A second plan...
 8.44: Aviation.A plane ew due north at 400 mph for 4 hours. A second plan...
 8.45: Aviation. A plane ew 30NW at 350mph for 2.5 hours. A second plane, ...
 8.46: Aviation. A plane ew 30NW at 350mph for 3 hours. A second plane sta...
 8.47: Baseball.A baseball diamond is actually a square that is 90 feet on...
 8.48: Aircraft Wing. Given the acute angle (14.4 ) and two sides (18 feet...
 8.49: Sliding Board. A 40foot slide leaning against the bottom of a buil...
 8.50: Airplane Slide. An airplane door is 6 feet high. If a slide attache...
 8.51: refer to the following: To quantify the torque (rotational force) o...
 8.52: refer to the following: To quantify the torque (rotational force) o...
 8.53: refer to the following: The term triangulation is often used to pin...
 8.54: refer to the following: The term triangulation is often used to pin...
 8.55: Law Enforcement. Two members of a SWAT team and the thief they are ...
 8.56: Law Enforcement. In reference to Exercise 55, what angle does the z...
 8.57: Rock Climbing. (See Exercise 44 in Section 8.1 for the context.) In...
 8.58: Bowling. Consider the following diagram:From the diagram, we know t...
 8.59: Bowling. Next, a bowler must convert a 78 spare and must hit the 8...
 8.60: Surveying. A glaciologist needs to determine the length across a ce...
 8.61: Surveying. A glaciologist needs to determine the length across a ce...
 8.62: Home Dcor. A table in the shape of a regular octagon is to be const...
 8.63: explain the mistake that is made. Solve the triangle and Solution: ...
 8.64: explain the mistake that is made. Solve the triangle . Solution: St...
 8.65: determine whether each statement is true or false.Given three sides...
 8.66: determine whether each statement is true or false.Given three angle...
 8.67: determine whether each statement is true or false.The Pythagorean t...
 8.68: determine whether each statement is true or false.The Law of Cosine...
 8.69: determine whether each statement is true or false.If an obtuse tria...
 8.70: determine whether each statement is true or false.All acute triangl...
 8.71: Show that . Hint: Apply the Law of Cosines.
 8.72: Show that Hint: Apply the Law of Cosines. a = c
 8.73: Consider the following diagram. Find
 8.74: Using the diagram in Exercise 73, Find tanaX 2b.
 8.75: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.76: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.77: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.78: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.79: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.80: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.1: nd the area of each triangle described.
 8.2: nd the area of each triangle described.
 8.3: nd the area of each triangle described.
 8.4: nd the area of each triangle described.
 8.5: nd the area of each triangle described.
 8.6: nd the area of each triangle described.
 8.7: nd the area of each triangle described.
 8.8: nd the area of each triangle described.
 8.9: nd the area of each triangle described.
 8.10: nd the area of each triangle described.
 8.11: nd the area of each triangle described.
 8.12: nd the area of each triangle described.
 8.13: nd the area of each triangle described.
 8.14: nd the area of each triangle described.
 8.15: nd the area of each triangle described.
 8.16: nd the area of each triangle described.
 8.17: nd the area of each triangle described.
 8.18: nd the area of each triangle described.
 8.19: nd the area of each triangle described.
 8.20: nd the area of each triangle described.
 8.21: nd the area of each triangle described.
 8.22: nd the area of each triangle described.
 8.23: nd the area of each triangle described.
 8.24: nd the area of each triangle described.
 8.25: nd the area of each triangle described.
 8.26: nd the area of each triangle described.
 8.27: nd the area of each triangle described.
 8.28: nd the area of each triangle described.
 8.29: nd the area of each triangle described.
 8.30: nd the area of each triangle described.
 8.31: nd the area of each triangle described.
 8.32: nd the area of each triangle described.
 8.33: Bermuda Triangle. Calculate the area of the so called Bermuda Trian...
 8.34: Bermuda Triangle. Calculate the area of the Bermuda Triangle, if, a...
 8.35: Triangular Tarp. A large triangular tarp is needed to cover a playg...
 8.36: Flower Seed. A triangular garden measures 41 feet 16 feet 28 feet. ...
 8.37: Mural. Some students are painting a mural on the side of a building...
 8.38: Mural. Some students are painting a mural on the side of a building...
 8.39: Insect Infestation. Some very destructive beetles have made their w...
 8.40: Real Estate. A real estate agent needs to determine the area of a t...
 8.41: refer to the following: A company is considering purchasing a trian...
 8.42: refer to the following: A company is considering purchasing a trian...
 8.43: Parking Lot. A parking lot is to have the shape of a parallelogram ...
 8.44: Parking Lot. A parking lot is to have the shape of a parallelogram ...
 8.45: Regular Hexagon. A regular hexagon has sides measuring 3 feet. What...
 8.46: Regular Decagon. A regular decagon has sides measuring 5 inches. Wh...
 8.47: Geometry. Three spokes of lengths 8, 6, and 11, respectively, emana...
 8.48: Construction. A rectangular great room, 15 ft 25 ft, has an open be...
 8.49: Geometry. A quadrilateral ABCD has sides of lengths AB 2, BC 3, CD ...
 8.50: Geometry. A quadrilateral ABCD has sides of lengths AB 5, BC 6, CD ...
 8.51: explain the mistake that is made. Calculate the area of the triangl...
 8.52: explain the mistake that is made. Calculate the area of the triangl...
 8.53: determine whether each statement is true or false.Herons formula ca...
 8.54: determine whether each statement is true or false. Herons formula c...
 8.55: determine whether each statement is true or false.If two triangles ...
 8.56: determine whether each statement is true or false. If two rhombi (i...
 8.57: determine whether each statement is true or false. If two parallelo...
 8.58: determine whether each statement is true or false.The semiperimeter...
 8.59: Show that the area for an SAA triangle is given byAssume that and a...
 8.60: Show that the area of an isosceles triangle with equal sides of len...
 8.61: Find the area of the shaded region.
 8.62: Find the area of the shaded region
 8.63: Prove that the area of a rhombus that has side length s with an ang...
 8.64: Prove that the area of a parallelogram with side lengths s and 3s a...
 8.65: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.66: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.67: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.68: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.69: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.70: let A, B, and C be the lengths of the three sides with X, Y, and Z ...
 8.1: nd the magnitude of the vector AB.
 8.2: nd the magnitude of the vector AB.
 8.3: nd the magnitude of the vector AB.
 8.4: nd the magnitude of the vector AB.
 8.5: nd the magnitude of the vector AB.
 8.6: nd the magnitude of the vector AB.
 8.7: nd the magnitude and direction angle of the given vector.
 8.8: nd the magnitude and direction angle of the given vector.
 8.9: nd the magnitude and direction angle of the given vector.
 8.10: nd the magnitude and direction angle of the given vector.
 8.11: nd the magnitude and direction angle of the given vector.
 8.12: nd the magnitude and direction angle of the given vector.
 8.13: nd the magnitude and direction angle of the given vector.
 8.14: nd the magnitude and direction angle of the given vector.
 8.15: nd the magnitude and direction angle of the given vector.
 8.16: nd the magnitude and direction angle of the given vector.
 8.17: perform the indicated vector operation, given and
 8.18: perform the indicated vector operation, given and
 8.19: perform the indicated vector operation, given and
 8.20: perform the indicated vector operation, given and
 8.21: perform the indicated vector operation, given and
 8.22: perform the indicated vector operation, given and
 8.23: perform the indicated vector operation, given and
 8.24: perform the indicated vector operation, given and
 8.25: nd the vector, given its magnitude and direction angle.
 8.26: nd the vector, given its magnitude and direction angle.
 8.27: nd the vector, given its magnitude and direction angle.
 8.28: nd the vector, given its magnitude and direction angle.
 8.29: nd the vector, given its magnitude and direction angle.
 8.30: nd the vector, given its magnitude and direction angle.
 8.31: nd the vector, given its magnitude and direction angle.
 8.32: nd the vector, given its magnitude and direction angle.
 8.33: nd the vector, given its magnitude and direction angle.
 8.34: nd the vector, given its magnitude and direction angle.
 8.35: nd a unit vector in the direction of the given vector.
 8.36: nd a unit vector in the direction of the given vector.
 8.37: nd a unit vector in the direction of the given vector.
 8.38: nd a unit vector in the direction of the given vector.
 8.39: nd a unit vector in the direction of the given vector.
 8.40: nd a unit vector in the direction of the given vector.
 8.41: nd a unit vector in the direction of the given vector.
 8.42: nd a unit vector in the direction of the given vector.
 8.43: nd a unit vector in the direction of the given vector.
 8.44: nd a unit vector in the direction of the given vector.
 8.45: express the vector in terms of unit vectors i and j.
 8.46: express the vector in terms of unit vectors i and j.
 8.47: express the vector in terms of unit vectors i and j.
 8.48: express the vector in terms of unit vectors i and j.
 8.49: express the vector in terms of unit vectors i and j.
 8.50: express the vector in terms of unit vectors i and j.
 8.51: perform the indicated vector operation.
 8.52: perform the indicated vector operation.
 8.53: perform the indicated vector operation.
 8.54: perform the indicated vector operation.
 8.55: perform the indicated vector operation.
 8.56: perform the indicated vector operation.
 8.57: Bullet Speed. A bullet is red from ground level at a speed of 2200 ...
 8.58: Weightlifting. A 50pound weight lies on an inclined bench that mak...
 8.59: Weight of a Boat. A force of 630 pounds is needed to pull a speedbo...
 8.60: Weight of a Boat. A force of 500 pounds is needed to pull a speedbo...
 8.61: Speed and Direction of a Ship. A ships captain sets a course due no...
 8.62: Speed and Direction of a Ship. A ships captain sets a course due we...
 8.63: Heading and Airspeed. A plane has a compass heading of east of due ...
 8.64: Heading and Airspeed. A plane has a compass heading of east of due ...
 8.65: Sliding Box. A box weighing 500 pounds is held in place on an incli...
 8.66: Sliding Box. A box weighing 500 pounds is held in place on an incli...
 8.67: Baseball. A baseball player throws a ball with an initial velocity ...
 8.68: Baseball. A baseball pitcher throws a ball with an initial velocity...
 8.69: refer to the following: In a post pattern in football, the receiver...
 8.70: refer to the following: In a post pattern in football, the receiver...
 8.71: Resultant Force. A force with a magnitude of 100 pounds and another...
 8.72: Resultant Force. A force with a magnitude of 100 pounds and another...
 8.73: Resultant Force. A force of 1000 pounds is acting on an object at a...
 8.74: Resultant Force. A force of 1000 pounds is acting on an object at a...
 8.75: Resultant Force. Forces with magnitudes of 200 N and 180 N act on a...
 8.76: Resultant Force. Forces with magnitudes of 100 N and 50 N act on a ...
 8.77: Exercise Equipment. A tether ball weighing 5 pounds is pulled outwa...
 8.78: Exercise Equipment. A tether ball weighing 8 pounds is pulled outwa...
 8.79: Recreation. A freshman wishes to sign up for four different clubs d...
 8.80: Recreation. A freshman wishes to sign up for three different clubs ...
 8.81: Torque. Torque is the tendency for an arm to rotate about a pivot p...
 8.82: Torque. You walk through a swinging mall door to enter a department...
 8.83: Torque. You walk through a swinging mall door to enter a department...
 8.84: Torque. Suppose that within the context of Exercises 82 and 83, the...
 8.85: Resultant Force. A person is walking two dogs fastened to separate ...
 8.86: Resultant Force. A person is walking three dogs fastened to separat...
 8.87: refer to the following: Muscle A and muscle B are attached to a bon...
 8.88: refer to the following: Muscle A and muscle B are attached to a bon...
 8.89: explain the mistake that is made. Find the magnitude of the vector ...
 8.90: explain the mistake that is made. Find the direction angle of the v...
 8.91: determine whether each statment is true or false.The magnitude of t...
 8.92: determine whether each statment is true or false.The arrow componen...
 8.93: determine whether each statment is true or false.The magnitude of a...
 8.94: determine whether each statment is true or false.The magnitude of a...
 8.95: Would a scalar or a vector represent the following? The car is driv...
 8.96: Would a scalar or vector represent the following? The granite has a...
 8.97: Find the magnitude of the vector if
 8.98: Find the direction angle of the vector if and b 7 0.
 8.99: Show that if is a unit vector in the direction of , then .
 8.100: Show that if is a unit vector, then (a, b) lies on the unit circle.
 8.101: A vector is a linear combination of and if there exist constants c1...
 8.102: Show that is a linear combination of and , for any real constants a...
 8.103: Prove that , showing carefully how all relevant properties and deni...
 8.104: Let . Compute
 8.105: refer to the following: Vectors can be represented as column matric...
 8.106: refer to the following: Vectors can be represented as column matric...
 8.107: refer to the following: Vectors can be represented as column matric...
 8.108: refer to the following: Vectors can be represented as column matric...
 8.109: refer to the following: Vectors can be represented as column matric...
 8.110: refer to the following: Vectors can be represented as column matric...
 8.1: nd the indicated dot product.
 8.2: nd the indicated dot product.
 8.3: nd the indicated dot product.
 8.4: nd the indicated dot product.
 8.5: nd the indicated dot product.
 8.6: nd the indicated dot product.
 8.7: nd the indicated dot product.
 8.8: nd the indicated dot product.
 8.9: nd the indicated dot product.
 8.10: nd the indicated dot product.
 8.11: nd the indicated dot product.
 8.12: nd the indicated dot product.
 8.13: nd the angle (round to the nearest degree) between each pair of vec...
 8.14: nd the angle (round to the nearest degree) between each pair of vec...
 8.15: nd the angle (round to the nearest degree) between each pair of vec...
 8.16: nd the angle (round to the nearest degree) between each pair of vec...
 8.17: nd the angle (round to the nearest degree) between each pair of vec...
 8.18: nd the angle (round to the nearest degree) between each pair of vec...
 8.19: nd the angle (round to the nearest degree) between each pair of vec...
 8.20: nd the angle (round to the nearest degree) between each pair of vec...
 8.21: nd the angle (round to the nearest degree) between each pair of vec...
 8.22: nd the angle (round to the nearest degree) between each pair of vec...
 8.23: nd the angle (round to the nearest degree) between each pair of vec...
 8.24: nd the angle (round to the nearest degree) between each pair of vec...
 8.25: determine whether each pair of vectors is orthogonal.
 8.26: determine whether each pair of vectors is orthogonal.
 8.27: determine whether each pair of vectors is orthogonal.
 8.28: determine whether each pair of vectors is orthogonal.
 8.29: determine whether each pair of vectors is orthogonal.
 8.30: determine whether each pair of vectors is orthogonal.
 8.31: determine whether each pair of vectors is orthogonal.
 8.32: determine whether each pair of vectors is orthogonal.
 8.33: determine whether each pair of vectors is orthogonal.
 8.34: determine whether each pair of vectors is orthogonal.
 8.35: determine whether each pair of vectors is orthogonal.
 8.36: determine whether each pair of vectors is orthogonal.
 8.37: Lifting Weights. How much work does it take to lift 100 pounds vert...
 8.38: Lifting Weights. How much work does it take to lift 150 pounds vert...
 8.39: Raising Wrecks. How much work is done by a crane to lift a 2ton ca...
 8.40: Raising Wrecks. How much work is done by a crane to lift a 2.5ton ...
 8.41: Work. To slide a crate across the oor, a force of 50 pounds at a an...
 8.42: Work. To slide a crate across the oor, a force of 800 pounds at a a...
 8.43: Close a Door. A sliding door is closed by pulling a cord with a con...
 8.44: Close a Door. A sliding door is closed by pulling a cord with a con...
 8.45: Braking Power. A car that weighs 2500 pounds is parked on a hill in...
 8.46: Towing Power. A car that weighs 2500 pounds is parked on a hill in ...
 8.47: Towing Power. A semi trailer truck that weighs 40,000 pounds is par...
 8.48: Braking Power. A truck that weighs 40,000 pounds is parked on a hil...
 8.49: Business. Supposethat representsthe number of units of battery A an...
 8.50: Demographics. Suppose that represents the number of males and femal...
 8.51: Geometry. Use vector methods to show that the diagonals of a rhombu...
 8.52: Geometry. Let be a unit vector, and consider the following diagram:...
 8.53: Geometry. Consider the following diagram:a. Compute cos, sin, cos, ...
 8.54: Geometry. Consider the diagram in Exercise 53. a. Compute cos, sin,...
 8.55: Tennis. A player hits an overhead smash at full arm extension at th...
 8.56: Tennis. In Exercise 55, use the dot product to determine the angle ...
 8.57: Optimization. Let be a given vector, and suppose that the head of l...
 8.58: Optimization. Let be a given vector, and suppose that the head of l...
 8.59: Pursuit Theory. Assume that the head of is restricted so that its t...
 8.60: Pursuit Theory. Assume that the head of is restricted so that its t...
 8.61: explain the mistake that is made.Find the dot product Solution: Mul...
 8.62: explain the mistake that is made.Find the dot product Solution: Mul...
 8.63: determine whether each statement is true or false.A dot product of ...
 8.64: determine whether each statement is true or false.A dot product of ...
 8.65: determine whether each statement is true or false.Orthogonal vector...
 8.66: determine whether each statement is true or false. If the dot produ...
 8.67: refer to the following to nd the dot product:The dot product of vec...
 8.68: refer to the following to nd the dot product:The dot product of vec...
 8.69: given and show that the following properties are true.u#v = v#u
 8.70: given and show that the following properties are true.u#u = u2
 8.71: given and show that the following properties are true.0#u = 0
 8.72: given and show that the following properties are true. k is a scalar
 8.73: Show that .
 8.74: Show that
 8.75: The projection of onto is dened by proj This vector is depicted bel...
 8.76: a. Compute proj b. What is proj for any c 0?
 8.77: Suppose that you are given a vector . For what vectors does proj
 8.78: True or False: proj proj proj
 8.79: If and are unit vectors, determine the maximum and minimum value of .
 8.80: Assume that the angle between and is . Show that
 8.81: nd the indicated dot product with a calculator.
 8.82: nd the indicated dot product with a calculator.
 8.83: A rectangle has sides with lengths 18 units and 11 units. Find the ...
 8.84: The denition of a dot product and the formula to nd the angle betwe...
 8.85: The denition of a dot product and the formula to nd the angle betwe...
 8.1: graph each complex number in the complex plane.
 8.2: graph each complex number in the complex plane.
 8.3: graph each complex number in the complex plane.
 8.4: graph each complex number in the complex plane.
 8.5: graph each complex number in the complex plane.
 8.6: graph each complex number in the complex plane.
 8.7: graph each complex number in the complex plane.
 8.8: graph each complex number in the complex plane.
 8.9: express each complex number in polar form.
 8.10: express each complex number in polar form.
 8.11: express each complex number in polar form.
 8.12: express each complex number in polar form.
 8.13: express each complex number in polar form.
 8.14: express each complex number in polar form.
 8.15: express each complex number in polar form.
 8.16: express each complex number in polar form.
 8.17: express each complex number in polar form.
 8.18: express each complex number in polar form.
 8.19: express each complex number in polar form.
 8.20: express each complex number in polar form.
 8.21: express each complex number in polar form.
 8.22: express each complex number in polar form.
 8.23: use a calculator to express each complex number in polar form.
 8.24: use a calculator to express each complex number in polar form.
 8.25: use a calculator to express each complex number in polar form.
 8.26: use a calculator to express each complex number in polar form.
 8.27: use a calculator to express each complex number in polar form.
 8.28: use a calculator to express each complex number in polar form.
 8.29: use a calculator to express each complex number in polar form.
 8.30: use a calculator to express each complex number in polar form.
 8.31: use a calculator to express each complex number in polar form.
 8.32: use a calculator to express each complex number in polar form.
 8.33: use a calculator to express each complex number in polar form.
 8.34: use a calculator to express each complex number in polar form.
 8.35: use a calculator to express each complex number in polar form.
 8.36: use a calculator to express each complex number in polar form.
 8.37: use a calculator to express each complex number in polar form.
 8.38: use a calculator to express each complex number in polar form.
 8.39: express each complex number in rectangular form.
 8.40: use a calculator to express each complex number in rectangular form.
 8.41: use a calculator to express each complex number in rectangular form.
 8.42: use a calculator to express each complex number in rectangular form.
 8.43: use a calculator to express each complex number in rectangular form.
 8.44: use a calculator to express each complex number in rectangular form.
 8.45: use a calculator to express each complex number in rectangular form.
 8.46: use a calculator to express each complex number in rectangular form.
 8.47: use a calculator to express each complex number in rectangular form.
 8.48: use a calculator to express each complex number in rectangular form.
 8.49: use a calculator to express each complex number in rectangular form.
 8.50: use a calculator to express each complex number in rectangular form.
 8.51: use a calculator to express each complex number in rectangular form.
 8.52: use a calculator to express each complex number in rectangular form.
 8.53: use a calculator to express each complex number in rectangular form.
 8.54: use a calculator to express each complex number in rectangular form.
 8.55: use a calculator to express each complex number in rectangular form.
 8.56: use a calculator to express each complex number in rectangular form.
 8.57: use a calculator to express each complex number in rectangular form.
 8.58: use a calculator to express each complex number in rectangular form.
 8.59: Resultant Force. Force A, at 100 pounds, and force B, at 120 pounds...
 8.60: Resultant Force. Force A, at 40 pounds, and force B, at 50 pounds, ...
 8.61: Resultant Force. Force A, at 80 pounds, and force B, at 150 pounds,...
 8.62: Resultant Force. Force A, at 20 pounds, and force B, at 60 pounds, ...
 8.63: Resultant Force. Refer to Section 8.4, Exercise 85. Express the vec...
 8.64: Resultant Force. Refer to Section 8.4, Exercise 86. Express the vec...
 8.65: Resultant Force.Refer to Section 8.4, Exercise 75. Express the resu...
 8.66: Resultant Force.Refer to Section 8.4, Exercise 76. Express the resu...
 8.67: explain the mistake that is made.Express in polar form. Solution: F...
 8.68: explain the mistake that is made.Express in polar form. Solution: F...
 8.69: determine whether each statement is true or false.In the complex pl...
 8.70: determine whether each statement is true or false. In the complex p...
 8.71: determine whether each statement is true or false.The modulus of z ...
 8.72: determine whether each statement is true or false.The argument of z...
 8.73: Find the argument of where a is a positive real number.
 8.74: Find the argument of where b is a positive real number.
 8.75: Find the modulus of where b is a negative real number.
 8.76: Find the modulus of where a is a negative real number.
 8.77: use a calculator to express the complex number in polar form. where
 8.78: use a calculator to express the complex number in polar form. where...
 8.79: Suppose that a complex number z lies on the circle x2 y2 2. If and ...
 8.80: Suppose that a complex number z lies on the circle x2 y2 8. If and ...
 8.81: Consider the following diagram:Find z in trigonometric form. (Hint:...
 8.82: Consider the following diagram:Find z in trigonometric form. (Hint:...
 8.83: Graphing calculators are able to convert complex numbers from recta...
 8.84: Graphing calculators are able to convert complex numbers from recta...
 8.85: A second way of using a graphing calculator to convert between rect...
 8.86: A second way of using a graphing calculator to convert between rect...
 8.87: Another way of using a graphing calculator to represent complex num...
 8.88: Another way of using a graphing calculator to represent complex num...
 8.1: nd the product z1z2 and express it in rectangular form. z1 = 4(cos ...
 8.2: nd the product z1z2 and express it in rectangular form. z1 = 2(cos1...
 8.3: nd the product z1z2 and express it in rectangular form. z1 = 4(cos ...
 8.4: nd the product z1z2 and express it in rectangular form. z1 = 3(cos ...
 8.5: nd the product z1z2 and express it in rectangular form. z1 = 2(cos ...
 8.6: nd the product z1z2 and express it in rectangular form. z1 = 3(cos ...
 8.7: nd the product z1z2 and express it in rectangular form. z1 = 13ccos...
 8.8: nd the product z1z2 and express it in rectangular form. z1 = 15ccos...
 8.9: nd the product z1z2 and express it in rectangular form. z1 = 4ccosa...
 8.10: nd the product z1z2 and express it in rectangular form. z1 = 6ccosa...
 8.11: nd the quotient and express it in rectangular form.z1 = 6(cos 100 +...
 8.12: nd the quotient and express it in rectangular form.z1 = 8(cos 80 + ...
 8.13: nd the quotient and express it in rectangular form.z1 = 10(cos 200 ...
 8.14: nd the quotient and express it in rectangular form.z1 = 4(cos 280 +...
 8.15: nd the quotient and express it in rectangular form.z1 = 112 (cos 35...
 8.16: nd the quotient and express it in rectangular form.z1 = 140 (cos 11...
 8.17: nd the quotient and express it in rectangular form.z1 = 9ccos a 5p ...
 8.18: nd the quotient and express it in rectangular form.z1 = 8ccosa5p 8b...
 8.19: nd the quotient and express it in rectangular form.z1 = 45ccosa22p ...
 8.20: nd the quotient and express it in rectangular form.z1 = 22ccosa11p ...
 8.21: nd the result of each expression using De Moivres theorem. Write th...
 8.22: nd the result of each expression using De Moivres theorem. Write th...
 8.23: nd the result of each expression using De Moivres theorem. Write th...
 8.24: nd the result of each expression using De Moivres theorem. Write th...
 8.25: nd the result of each expression using De Moivres theorem. Write th...
 8.26: nd the result of each expression using De Moivres theorem. Write th...
 8.27: nd the result of each expression using De Moivres theorem. Write th...
 8.28: nd the result of each expression using De Moivres theorem. Write th...
 8.29: nd the result of each expression using De Moivres theorem. Write th...
 8.30: nd the result of each expression using De Moivres theorem. Write th...
 8.31: nd all nth roots of z. Write the answers in polar form, and plot th...
 8.32: nd all nth roots of z. Write the answers in polar form, and plot th...
 8.33: nd all nth roots of z. Write the answers in polar form, and plot th...
 8.34: nd all nth roots of z. Write the answers in polar form, and plot th...
 8.35: nd all nth roots of z. Write the answers in polar form, and plot th...
 8.36: nd all nth roots of z. Write the answers in polar form, and plot th...
 8.37: nd all nth roots of z. Write the answers in polar form, and plot th...
 8.38: nd all nth roots of z. Write the answers in polar form, and plot th...
 8.39: nd all nth roots of z. Write the answers in polar form, and plot th...
 8.40: nd all nth roots of z. Write the answers in polar form, and plot th...
 8.41: nd all complex solutions to the given equations.
 8.42: nd all complex solutions to the given equations.
 8.43: nd all complex solutions to the given equations.
 8.44: nd all complex solutions to the given equations.
 8.45: nd all complex solutions to the given equations.
 8.46: nd all complex solutions to the given equations.
 8.47: nd all complex solutions to the given equations.
 8.48: nd all complex solutions to the given equations.
 8.49: nd all complex solutions to the given equations.
 8.50: nd all complex solutions to the given equations.
 8.51: nd all complex solutions to the given equations.
 8.52: nd all complex solutions to the given equations.
 8.53: nd all complex solutions to the given equations.
 8.54: nd all complex solutions to the given equations.
 8.55: nd all complex solutions to the given equations.
 8.56: nd all complex solutions to the given equations.
 8.57: Complex Pentagon. When you graph the ve fth roots ofand connect the...
 8.58: Complex Square. When you graph the four fourth roots of 16i and con...
 8.59: Hexagon. Compute the six sixth roots of , and form a hexagon by con...
 8.60: Octagon. Compute the eight eighth roots of 2i, and form an octagon ...
 8.61: explain the mistake that is made. Let and Find Solution: Use the qu...
 8.62: explain the mistake that is made. Let and Find Solution: Write the ...
 8.63: explain the mistake that is made. Find Solution: Raise each term to...
 8.64: explain the mistake that is made. Find all complex solutions to Sol...
 8.65: determine whether the statement is true or false.The product of two...
 8.66: determine whether the statement is true or false.The quotient of tw...
 8.67: determine whether the statement is true or false.There are always n...
 8.68: determine whether the statement is true or false.There are always n...
 8.69: determine whether the statement is true or false.There are n distin...
 8.70: determine whether the statement is true or false.There exists a com...
 8.71: use the following identity: In calculus you will see an identity ca...
 8.72: use the following identity: In calculus you will see an identity ca...
 8.73: use the following identity: In calculus you will see an identity ca...
 8.74: use the following identity: In calculus you will see an identity ca...
 8.75: Use De Moivres theorem to prove the identity
 8.76: Use De Moivres theorem to derive an expression for
 8.77: Use De Moivres theorem to derive an expression for
 8.78: Calculate a1 2 +13 2ib14a1 2 13 2ib
 8.79: Calculate where n and m are positive integers.
 8.80: Calculate where n and m are positive integers.
 8.81: refer to the following: According to the nth root theorem, the rst ...
 8.82: refer to the following: According to the nth root theorem, the rst ...
 8.83: refer to the following: According to the nth root theorem, the rst ...
 8.84: refer to the following: According to the nth root theorem, the rst ...
 8.85: refer to the following: According to the nth root theorem, the rst ...
 8.86: refer to the following: According to the nth root theorem, the rst ...
 8.1: plot each indicated point in a polar coordinate system.
 8.2: plot each indicated point in a polar coordinate system.
 8.3: plot each indicated point in a polar coordinate system.
 8.4: plot each indicated point in a polar coordinate system.
 8.5: plot each indicated point in a polar coordinate system.
 8.6: plot each indicated point in a polar coordinate system.
 8.7: plot each indicated point in a polar coordinate system.
 8.8: plot each indicated point in a polar coordinate system.
 8.9: plot each indicated point in a polar coordinate system.
 8.10: plot each indicated point in a polar coordinate system.
 8.11: convert each point to exact polar coordinates. Assume that 0 2.
 8.12: convert each point to exact polar coordinates. Assume that 0 2.
 8.13: convert each point to exact polar coordinates. Assume that 0 2.
 8.14: convert each point to exact polar coordinates. Assume that 0 2.
 8.15: convert each point to exact polar coordinates. Assume that 0 2.
 8.16: convert each point to exact polar coordinates. Assume that 0 2.
 8.17: convert each point to exact polar coordinates. Assume that 0 2.
 8.18: convert each point to exact polar coordinates. Assume that 0 2.
 8.19: convert each point to exact polar coordinates. Assume that 0 2.
 8.20: convert each point to exact polar coordinates. Assume that 0 2.
 8.21: convert each point to exact rectangular coordinates.
 8.22: convert each point to exact rectangular coordinates.
 8.23: convert each point to exact rectangular coordinates.
 8.24: convert each point to exact rectangular coordinates.
 8.25: convert each point to exact rectangular coordinates.
 8.26: convert each point to exact rectangular coordinates.
 8.27: convert each point to exact rectangular coordinates.
 8.28: convert each point to exact rectangular coordinates.
 8.29: convert each point to exact rectangular coordinates.
 8.30: convert each point to exact rectangular coordinates.
 8.31: match the polar graphs with their corresponding equations.
 8.32: match the polar graphs with their corresponding equations.
 8.33: match the polar graphs with their corresponding equations.
 8.34: match the polar graphs with their corresponding equations.
 8.35: graph each equation.
 8.36: graph each equation.
 8.37: graph each equation.
 8.38: graph each equation.
 8.39: graph each equation.
 8.40: graph each equation.
 8.41: graph each equation.
 8.42: graph each equation.
 8.43: graph each equation.
 8.44: graph each equation.
 8.45: graph each equation.
 8.46: graph each equation.
 8.47: graph each equation.
 8.48: graph each equation.
 8.49: graph each equation.
 8.50: graph each equation.
 8.51: convert the equation from polar to rectangular form. Identify the r...
 8.52: convert the equation from polar to rectangular form. Identify the r...
 8.53: convert the equation from polar to rectangular form. Identify the r...
 8.54: convert the equation from polar to rectangular form. Identify the r...
 8.55: graph the polar equation.
 8.56: graph the polar equation.
 8.57: graph the polar equation.
 8.58: graph the polar equation.
 8.59: graph the polar equation.
 8.60: graph the polar equation.
 8.61: Halleys Comet. Halleys comet travels an elliptical path that can be...
 8.62: Dwarf Planet Pluto. The planet Pluto travels in an elliptical orbit...
 8.63: Archimedes Spiral. Spirals are seen in nature, as in the swirl of a...
 8.64: Archimedes Spiral. Spirals are seen in nature, as in the swirl of a...
 8.65: refer to the following: The lemniscate motion occurs naturally in t...
 8.66: refer to the following: The lemniscate motion occurs naturally in t...
 8.67: refer to the following: Many microphone manufacturers advertise tha...
 8.68: refer to the following: Many microphone manufacturers advertise tha...
 8.69: refer to the following: The sword artistry of the Samurai is legend...
 8.70: refer to the following: The sword artistry of the Samurai is legend...
 8.71: Home Improvement. The owner of a garden maze has decided to replace...
 8.72: Home Improvement. Consider the garden maze in Exercise 71, but now ...
 8.73: Magnetic Pendulum. A magnetic bob is afxed to an arm of length L, w...
 8.74: Magnetic Pendulum. In reference to the context of Exercise 73, now ...
 8.75: explain the mistake that is made.Convert to polar coordinates. Solu...
 8.76: explain the mistake that is made.Convert to polar coordinates. Solu...
 8.77: determine whether each statement is true or false.All cardioids are...
 8.78: determine whether each statement is true or false.All limaons are c...
 8.79: Find the polar equation that is equivalent to a vertical line, x = a.
 8.80: Find the polar equation that is equivalent to a horizontal line,
 8.81: Give another pair of polar coordinates for the point
 8.82: Convert to polar coordinates. Assume that a 7 0, b 7 0.
 8.83: Determine the values of at which r 4cos and rcos 1 intersect. Graph...
 8.84: Find the Cartesian equation for r asin bcos, where a and b are posi...
 8.85: Find the Cartesian equation for r =a sin(2 u) cos3 u  sin3 u
 8.86: Identify an equation for the following graph:
 8.87: Consider the equation r 2acos( b). Sketch the graph for various val...
 8.88: Consider the equation r asin(b), where a, b 0. Determine the smalle...
 8.89: Given , nd the for the inner loopabove the xaxis.
 8.90: Given , nd the for the petal inthe rst quadrant.
 8.91: Given nd the for the inner loop. uintervalsr = 1
 8.92: Given and nd all points of intersection.
 8.93: Given r 2 sin(4) and r 1, nd the angles of all the points of inters...
 8.94: Given r 2 cos(3) and r 1.5, nd the angles of all points of intersec...
 8.1: Solve the given triangles. a = 10, b = 20, a = 4
 8.2: Solve the given triangles. b = 40, g = 60, b = 10
 8.3: Solve the given triangles. a = 5, b = 45, c = 10
 8.4: Solve the given triangles. b = 60, g = 70, a = 20
 8.5: Solve the given triangles. g = 11, a = 11, c = 11
 8.6: Solve the given triangles. b = 20, g = 50, b = 8
 8.7: Solve the given triangles. a = 45, g = 45, b = 2
 8.8: Solve the given triangles. a = 60, b = 20, c = 17
 8.9: Solve the given triangles. a = 12, g = 22, a = 99
 8.10: Solve the given triangles. b = 102, g = 27, a = 24
 8.11: Two sides and an angle are given. Determine whether a triangle (or ...
 8.12: Two sides and an angle are given. Determine whether a triangle (or ...
 8.13: Two sides and an angle are given. Determine whether a triangle (or ...
 8.14: Two sides and an angle are given. Determine whether a triangle (or ...
 8.15: Two sides and an angle are given. Determine whether a triangle (or ...
 8.16: Two sides and an angle are given. Determine whether a triangle (or ...
 8.17: Two sides and an angle are given. Determine whether a triangle (or ...
 8.18: Two sides and an angle are given. Determine whether a triangle (or ...
 8.19: How Far from Home? Gary walked 8 miles due north, made a turn to th...
 8.20: How Far from Home? Mary walked 10 miles due north, made a turn to t...
 8.21: Solve each triangle.
 8.22: Solve each triangle.
 8.23: Solve each triangle.
 8.24: Solve each triangle.
 8.25: Solve each triangle.
 8.26: Solve each triangle.
 8.27: Solve each triangle.
 8.28: Solve each triangle.
 8.29: Solve each triangle.
 8.30: Solve each triangle.
 8.31: Solve each triangle.
 8.32: Solve each triangle.
 8.33: Solve each triangle.
 8.34: Solve each triangle.
 8.35: Solve each triangle.
 8.36: Solve each triangle.
 8.37: Solve each triangle.
 8.38: Solve each triangle.
 8.39: Solve each triangle.
 8.40: Solve each triangle.
 8.41: Find the area of each triangle described.
 8.42: Find the area of each triangle described.
 8.43: Find the area of each triangle described.
 8.44: Find the area of each triangle described.
 8.45: Find the area of each triangle described.
 8.46: Find the area of each triangle described.
 8.47: Find the area of each triangle described.
 8.48: Find the area of each triangle described.
 8.49: Area of Inscribed Triangle. The area of a triangle inscribed in a c...
 8.50: Area of Inscribed Triangle. The area of a triangle inscribed in a c...
 8.51: Find the magnitude of vector AB z. A = (4, 3) and B = (8, 2)
 8.52: Find the magnitude of vector AB z. A = (2, 11) and B = (2, 8)
 8.53: Find the magnitude of vector AB z. A = (0, 3) and B = (5, 9)
 8.54: Find the magnitude of vector AB z. v
 8.55: Find the magnitude and direction angle of the given vector
 8.56: Find the magnitude and direction angle of the given vector
 8.57: Find the magnitude and direction angle of the given vector
 8.58: Find the magnitude and direction angle of the given vector
 8.59: Perform the vector operation, given that and
 8.60: Perform the vector operation, given that and
 8.61: Find a unit vector in the direction of the given vector.
 8.62: Perform the indicated vector operation.
 8.63: Find the indicated dot product.
 8.64: Find the indicated dot product.
 8.65: Find the indicated dot product.
 8.66: Find the indicated dot product.
 8.67: Find the indicated dot product.
 8.68: Find the indicated dot product.
 8.69: Find the indicated dot product.
 8.70: Find the indicated dot product.
 8.71: Find the indicated dot product.
 8.72: Find the indicated dot product.
 8.73: Find the indicated dot product.
 8.74: Find the indicated dot product.
 8.75: Find the indicated dot product.
 8.76: Find the indicated dot product.
 8.77: Find the angle (round to the nearest degree) between each pair of v...
 8.78: Find the angle (round to the nearest degree) between each pair of v...
 8.79: Find the angle (round to the nearest degree) between each pair of v...
 8.80: Find the angle (round to the nearest degree) between each pair of v...
 8.81: Find the angle (round to the nearest degree) between each pair of v...
 8.82: Find the angle (round to the nearest degree) between each pair of v...
 8.83: Determine whether each pair of vectors is orthogonal.
 8.84: Determine whether each pair of vectors is orthogonal.
 8.85: Determine whether each pair of vectors is orthogonal.
 8.86: Determine whether each pair of vectors is orthogonal.
 8.87: Determine whether each pair of vectors is orthogonal.
 8.88: Determine whether each pair of vectors is orthogonal.
 8.89: Determine whether each pair of vectors is orthogonal.
 8.90: Determine whether each pair of vectors is orthogonal.
 8.91: Graph each complex number in the complex plane.
 8.92: Graph each complex number in the complex plane.
 8.93: Express each complex number in polar form.
 8.94: Express each complex number in polar form.
 8.95: Express each complex number in polar form.
 8.96: Express each complex number in polar form.
 8.97: With a calculator, express each complex number in polar form.
 8.98: With a calculator, express each complex number in polar form.
 8.99: With a calculator, express each complex number in polar form.
 8.100: With a calculator, express each complex number in polar form.
 8.101: Express each complex number in rectangular form.
 8.102: Express each complex number in rectangular form.
 8.103: Express each complex number in rectangular form.
 8.104: Express each complex number in rectangular form.
 8.105: With a calculator, express each complex number in rectangular form....
 8.106: With a calculator, express each complex number in rectangular form....
 8.107: Find the product 3(cos 200 + i sin 200) and 4(cos 70 + i sin 70)
 8.108: Find the product
 8.109: Find the product 3(cos 20 + i sin 20) and 4(cos 220 + i sin 220)
 8.110: Find the product cos 290 + i sin 290) and 4(cos 40 + i sin 40)
 8.111: 16(cos 200 + i sin 200) and 16(cos 50 + i sin 50) 16(cos 200 + i si...
 8.112: 16(cos 200 + i sin 200) and 16(cos 50 + i sin 50) 18(cos 190 + i si...
 8.113: 16(cos 200 + i sin 200) and 16(cos 50 + i sin 50) 24(cos 290 + i si...
 8.114: 16(cos 200 + i sin 200) and 16(cos 50 + i sin 50) 1200(cos 93 + i s...
 8.115: Find the result of each expression using De Moivres theorem. Write ...
 8.116: Find the result of each expression using De Moivres theorem. Write ...
 8.117: Find the result of each expression using De Moivres theorem. Write ...
 8.118: Find the result of each expression using De Moivres theorem. Write ...
 8.119: Find all nth roots of z. Write the answers in polar form, and plot ...
 8.120: Find all nth roots of z. Write the answers in polar form, and plot ...
 8.121: Find all nth roots of z. Write the answers in polar form, and plot ...
 8.122: Find all nth roots of z. Write the answers in polar form, and plot ...
 8.123: Find all complex solutions to the given equations
 8.124: Find all complex solutions to the given equations
 8.125: Find all complex solutions to the given equations
 8.126: Find all complex solutions to the given equations
 8.127: Convert each point to exact polar coordinates (assuming that and th...
 8.128: Convert each point to exact polar coordinates (assuming that and th...
 8.129: Convert each point to exact polar coordinates (assuming that and th...
 8.130: Convert each point to exact polar coordinates (assuming that and th...
 8.131: Convert each point to exact polar coordinates (assuming that and th...
 8.132: Convert each point to exact polar coordinates (assuming that and th...
 8.133: Convert each point to exact polar coordinates (assuming that and th...
 8.134: Convert each point to exact rectangular coordinates.
 8.135: Convert each point to exact rectangular coordinates.
 8.136: Convert each point to exact rectangular coordinates.
 8.137: Convert each point to exact rectangular coordinates.
 8.138: Convert each point to exact rectangular coordinates.
 8.139: Graph each equation.
 8.140: Graph each equation.
 8.141: Graph each equation.
 8.142: Graph each equation.
 8.143: Graph each equation.
 8.144: Graph each equation.
 8.145: Let A, B, and C be the lengths of the three sides of a triangle wit...
 8.146: Let A, B, and C be the lengths of the three sides of a triangle wit...
 8.147: With the graphing calculator SUM command, nd the magnitude of the g...
 8.148: Another way of using a graphing calculator to represent complex num...
 8.149: Write in polar form using the SUM command to nd its modulus, and ro...
 8.150: Write in polar form using the SUM command to nd its modulus, and ro...
 8.151: Find the fourth roots of , and draw the complex rectangle with the ...
 8.152: Find the fourth roots of , and draw the complex rectangle with the ...
 8.153: Another way of using a graphing calculator to represent complex num...
 8.154: Another way of using a graphing calculator to represent complex num...
 8.155: Find the fourth roots of , and draw the complex rectangle with the ...
 8.156: Find the fourth roots of , and draw the complex rectangle with the ...
 8.157: Given nd the angles of all points of intersection (where
 8.158: Given nd the angles of all points of intersection (where .
 8.1: Solve the triangles, if possible.a = 30, b = 40, b = 10
 8.2: Solve the triangles, if possible.a = 47, b = 98, g = 35
 8.3: Solve the triangles, if possible.a = 7, b = 9, c = 12
 8.4: Solve the triangles, if possible.
 8.5: Solve the triangles, if possible.
 8.6: Solve the triangles, if possible.
 8.7: Solve the triangles, if possible.
 8.8: Solve the triangles, if possible.
 8.9: nd the areas of the given triangles.g = 72, a = 10, b = 12
 8.10: nd the areas of the given triangles.a = 7, b = 10, c = 13
 8.11: Find the magnitude and direction angle of the vector
 8.12: Find a unit vector pointing in the same direction as
 8.13: Perform the indicated operations: a.
 8.14: In a post pattern in football, the receiver in motion runs past the...
 8.15: use the complex number
 8.16: Find
 8.17: Find the four distinct fourth roots of z.
 8.18: Convert the point to rectangular coordinates.
 8.19: True or False: If is perpendicular to then . 2
 8.20: Solve
 8.21: Convert to a Cartesian equation:
 8.22: With the graphing calculator SUM command, nd the angle (round to th...
 8.1: Solve using the square root method:
 8.2: Solve for x, and express the solution in interval notation:
 8.3: Find the slope of the line that passes through the points
 8.4: Transform the equation into standard form by completing the square....
 8.5: Using the function evaluate the differencequotient
 8.6: Using the function evaluate the differencequotient Given the piecew...
 8.7: Using the function evaluate the differencequotient"e. State the dom...
 8.8: Find a quadratic function that has the vertex (1.5, 2.5) and point ...
 8.9: List the possible rational zeros, and test to determine all rationa...
 8.10: List the possible rational zeros, and test to determine all rationa...
 8.11: Graph the rational function Give all asymptotes.
 8.12: Graph the exponential function and state the yintercept, the domai...
 8.13: How much money should be put in a savings account now that earns 5....
 8.14: Graph the function using transformation techniques.
 8.15: Use properties of logarithms to simplify the expression
 8.16: Solve for x: ln
 8.17: In a 45 45 90 triangle, if the two legs have length 15 feet, how ...
 8.18: Find the positive measure of (rounded to the nearest degree) when t...
 8.19: Graph over one period.
 8.20: If and the terminal side of lies in quadrant II, nd the exact value...
 8.21: Find the exact value of when , cos a = 1 5cos( a  b) u usin u = 8 ...
 8.22: Solve the trigonometric equation exactly, over the interval
 8.23: Solve the triangle with .
 8.24: Given and , nd
 8.25: Express the complex number
 8.26: in rectangular form.
 8.27: With the graphing calculator command, nd the magnitude of vector an...
Solutions for Chapter 8: Additional Topics in Trigonometry
Full solutions for Algebra and Trigonometry  3rd Edition
ISBN: 9780470648032
Solutions for Chapter 8: Additional Topics in Trigonometry
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CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.