Measuring Distance Indirectly Juan wants to find the

In Exercises 1-4, find an angle between \(0^{\circ}\) and \(180^{\circ}\) that is a solution to the equation. cos A = 3/5

In Exercises 1-4, find an angle between \(0^{\circ}\) and \(180^{\circ}\) that is a solution to the equation. cos C = -0.23

In Exercises 1-4, find an angle between \(0^{\circ}\) and \(180^{\circ}\) that is a solution to the equation. cos A = -0.68

In Exercises 1-4, find an angle between \(0^{\circ}\) and \(180^{\circ}\) that is a solution to the equation. 3 cos C = 1.92

In Exercises 5 and 6, solve the equation (in terms of x and y) for (a) cos A and (b) A, \(0 \leq A \leq 180^{\circ}\). \(9^{2}=x^{2}+y^{2}-2 x y \cos A\)

In Exercises 5 and 6, solve the equation (in terms of x and y) for (a) cos A and (b) A, \(0 \leq A \leq 180^{\circ}\). \(y^{2}=x^{2}+25-10 \cos A\)

In Exercises 7-10, find a quadratic polynomial with real coefficients that satisfies the given condition. Has two positive zeros

In Exercises 7-10, find a quadratic polynomial with real coefficients that satisfies the given condition. Has one positive and one negative zero

In Exercises 7-10, find a quadratic polynomial with real coefficients that satisfies the given condition. Has no real zeros

In Exercises 7-10, find a quadratic polynomial with real coefficients that satisfies the given condition. Has exactly one positive zero

In Exercises 1-4, solve the triangle.

In Exercises 1-4, solve the triangle.

In Exercises 1-4, solve the triangle.

In Exercises 1-4, solve the triangle.

In Exercises 5-16, solve the triangle. \(A = 55^{\circ}\), b = 12, c = 7

In Exercises 5-16, solve the triangle. \(B = 35^{\circ}\), a = 43, c = 19

In Exercises 5-16, solve the triangle. a = 12, b = 21, \(C = 95^{\circ}\)

In Exercises 5-16, solve the triangle. b = 22, c = 31, \(A = 82^{\circ}\)

In Exercises 5-16, solve the triangle. a = 1, b = 5, c = 4

In Exercises 5-16, solve the triangle. a = 1, b = 5, c = 8

In Exercises 5-16, solve the triangle. a = 3.2, b = 7.6, c = 6.4

In Exercises 5-16, solve the triangle. a = 9.8, b = 12, c = 23

In Exercises 5-16, solve the triangle. \(A = 42^{\circ}\), a = 7, b = 10

In Exercises 5-16, solve the triangle. \(A = 57^{\circ}\), a = 11, b = 10

In Exercises 5-16, solve the triangle. \(A = 63^{\circ}\), a = 8.6, b = 11.1

In Exercises 5-16, solve the triangle. \(A = 71^{\circ}\), a = 9.3, b = 8.5

In Exercises 17-20, find the area of the triangle. \(A = 47^{\circ}\), b = 32 ft, c = 19 ft

In Exercises 17-20, find the area of the triangle. \(A = 52^{\circ}\), b = 14 m, c = 21 m

In Exercises 17-20, find the area of the triangle. \(B = 101^{\circ}\), a = 10 cm, c = 22 cm

In Exercises 17-20, find the area of the triangle. \(C = 112^{\circ}\), a = 1.8 in., b = 5.1 in.

In Exercises 21-28, decide whether a triangle can be formed with the given side lengths. If so, use Heron's Formula to find the area of the triangle. a = 4, b = 5, c = 8

In Exercises 21-28, decide whether a triangle can be formed with the given side lengths. If so, use Heron's Formula to find the area of the triangle. a = 5, b = 9, c = 7

In Exercises 21-28, decide whether a triangle can be formed with the given side lengths. If so, use Heron's Formula to find the area of the triangle. a = 3, b = 5, c = 8

In Exercises 21-28, decide whether a triangle can be formed with the given side lengths. If so, use Heron's Formula to find the area of the triangle. a = 23, b = 19, c = 12

In Exercises 21-28, decide whether a triangle can be formed with the given side lengths. If so, use Heron's Formula to find the area of the triangle. a = 19.3, b = 22.5, c = 31

In Exercises 21-28, decide whether a triangle can be formed with the given side lengths. If so, use Heron's Formula to find the area of the triangle. a = 8.2, b = 12.5, c = 28

In Exercises 21-28, decide whether a triangle can be formed with the given side lengths. If so, use Heron's Formula to find the area of the triangle. a = 33.4, b = 28.5, c = 22.3

In Exercises 21-28, decide whether a triangle can be formed with the given side lengths. If so, use Heron's Formula to find the area of the triangle. a = 18.2, b = 17.1, c = 12.3

Find the radian measure of the largest angle in the triangle with sides of 4, 5, and 6.

A parallelogram has sides of 18 and 26 ft, and an angle of \(39^{\circ}\). Find the shorter diagonal.

Find the area of a regular hexagon inscribed in a circle of radius 12 inches.

Find the area of a regular nonagon (9 sides) inscribed in a circle of radius 10 inches.

Find the area of a regular hexagon circumscribed about a circle of radius 12 inches. [Hint: Start by finding the distance from a vertex of the hexagon to the center of the circle.]

Find the area of a regular nonagon (9 sides) circumscribed about a circle of radius 10 inches.

Measuring Distance Indirectly Juan wants to find the distance between two points A and B on opposite sides of a building. He locates a point C that is 110 ft from A and 160 ft from B, as illustrated in the figure. If the angle at C is \(54^{\circ}\), find distance AB.

Designing a Baseball Field (a) Find the distance from the center of the front edge of the pitcher's rubber to the far corner of second base. How does this distance compare with the distance from the pitcher's rubber to first base? (See Example 5.) (b) Find \(\angle B\) in \(\Delta A B C\).

Designing a Softball Field In softball, adjacent bases are 60 ft apart. The distance from the center of the front edge of the pitcher's rubber to the far corner of home plate is 40 ft. (a) Find the distance from the center of the pitcher's rubber to the far corner of first base. (b) Find the distance from the center of the pitcher's rubber to the far corner of second base. (c) Find \(\angle B\) in \(\Delta A B C\).

Surveyors Calculations Tony must find the distance from A to B on opposite sides of a lake. He locates a point C that is 860 ft from A and 175 ft from B. He measures the angle at C to be \(78^{\circ}\). Find distance AB.

Construction Engineering A manufacturer is designing the roof truss that is modeled in the figure shown. (a) Find the measure of \(\angle C A E\). (b) If AF = 12 ft, find the length DF. (c) Find the length EF

Navigation Two airplanes flying together in formation take off in different directions. One flies due east at 350 mph, and the other flies east-northeast at 380 mph. How far apart are the two airplanes 2 hr after they separate, assuming that they fly at the same altitude?

Football Kick The player waiting to receive a kickoff stands at the 5 yard line (point A) as the ball is being kicked 65 yd up the field from the opponents 30 yard line. The kicked ball travels 73 yd at an angle of \(8^{\circ}\) to the right of the receiver, as shown in the figure (point B). Find the distance the receiver runs to catch the ball.

Architectural Design Building Inspector Julie Wang checks a building in the shape of a regular octagon, each side 20 ft long. She checks that the contractor has located the corners of the foundation correctly by measuring several of the diagonals. Calculate what the lengths of diagonals HB, HC, and HD should be.

Connecting Trigonometry and Geometry \(\angle C A B\) is inscribed in a rectangular box whose sides are 1, 2, and 3 ft long as shown. Find the measure of \(\angle C A B\).

Connecting Trigonometry and Geometry A cube has edges of length 2 ft. Point A is the midpoint of an edge. Find the measure of \(\angle A B C\).

True or False If \(\triangle A B C\) is any triangle with sides and angles labeled in the usual way, then \(b^{2}+c^{2}>2 b c \cos A\). Justify your answer.

True or False If a, b, and \(\theta\) are two sides and an included angle of a parallelogram, the area of the parallelogram is \(a b \sin \theta\). Justify your answer.

What is the area of a regular dodecagon (12-sided figure) inscribed in a circle of radius 12? (A) 427 (B) 432 (C) 437 (D) 442 (E) 447

The area of a triangle with sides 7, 8, and 9 is (A) \(6 \sqrt{15}\). (B) \(12 \sqrt{5}\). (C) \(16 \sqrt{3}\). (D) \(17 \sqrt{3}\). (E) \(18 \sqrt{3}\).

Two boats start at the same point and speed away along courses that form a \(110^{\circ}\) angle. If one boat travels at 24 miles per hour and the other boat travels at 32 miles per hour, how far apart are the boats after 30 minutes? (A) 21 miles (B) 22 miles (C) 23 miles (D) 24 miles (E) 25 miles

What is the measure of the smallest angle in a triangle with sides 12, 17, and 25? (A) \(21^{\circ}\) (B) \(22^{\circ}\) (C) \(23^{\circ}\) (D) \(24^{\circ}\) (E) \(25^{\circ}\)

Find the area of a regular polygon with n sides inscribed inside a circle of radius r. (Express your answer in terms of n and r.)

(a) Prove the identity: \(\frac{\cos A}{a}=\frac{b^{2}+c^{2}-a^{2}}{2 a b c}\). (b) Prove the (tougher) identity: \(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\frac{a^{2}+b^{2}+c^{2}}{2 a b c}\) [Hint: Use the identity in part (a), along with its other variations.]

Two ships leave a common port at 8:00 A.M. and travel at a constant rate of speed. Each ship keeps a log showing its distance from port and its distance from the other ship. Portions of the logs from later that morning for both ships are shown in the following tables. (a) How fast is each ship traveling? (Express your answer in knots, which are nautical miles per hour.) (b) What is the angle of intersection of the courses of the two ships? (c) How far apart are the ships at 12:00 noon if they maintain the same courses and speeds?

Prove that the area of a triangle can be found with the formula \(\triangle \text { Area }=\frac{a^{2} \sin B \sin C}{2 \sin A}\).

A segment of a circle is the region enclosed between a chord of a circle and the arc intercepted by the chord. Find the area of a segment intercepted by a 7-inch chord in a circle of radius 5 inches.