College Trigonometry MATH 124
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This 17 page Class Notes was uploaded by Lisa Wisoky on Monday October 12, 2015. The Class Notes belongs to MATH 124 at Fayetteville State University taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/221585/math-124-fayetteville-state-university in Mathematics (M) at Fayetteville State University.
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Date Created: 10/12/15
Chapter 4 Applications of Trigonometric Functions 41 Right triangle trigonometry Applications 1 A triangle in which one angle is a right angle 900 is called a The side opposite the right angle is called the and the remaining two sides are called the 2 Pythagorean Theorem 3 16isan ie00lt6lt900or0lt6lt 2 Place 6 in standard position then the coordinates of the point P are 3 P is a point on the terminal side of 6 that is also on the circle 4 We can express the trigonometric functions of 6 as ratios of the sides of a right triangle sin6 csc6 cos6 sec6 138116 cot6 4 Find the exact value of the siX trigonometric functions of the angle 6 in a right triangle with hypotenuse 5 and adjacent 3 5 1 If the sum of two angles are a right angle we say that these two angle are 2 Side a adjacent to 6 and opposite oz side I opposite 6 and adjacent to oz 3 Cofunctions sina cosa tenor csca seca cota Because of these relations the functions sine and cosine tangent and cotangent and secant and cosecant are called of each other Recall sing 6 cos 6 cos 6 sin 6 6 Theorem Cofunctions of complementary angles are equal Example sin 300 cos 600 ten 450 cot 450 sec 800 csc 100 7 Example Simply the following expressions 57r 1 sin cos 12 2 sin2 400 sin2 500 3 tan 200 tan 700 8 To solve a right triangle means to nd the missing lengths of its sides and the measurements of its angles For the above right triangle we have 8 Example If I 2 and oz 400 in a right triangle nd a c and 6 a m168c 261 9 Example If a 3 and b 2 in a right triangle nd 0 oz and 6 oz 5630 10 Example A surveyor can measure the width of a river by setting up a transit at a point C on one side of the river and taking a sighting of a point A on the other side After turning an angle of 900 at C the surveyor walks a distance of 200 meters to point B Using the transit at B the angle 6 is measured and found to be 200 What is the width of the river I m 7279 42 The laws of sines 1 If none of the angles of a triangle is a right angle the triangle is called 2 Notation We label an oblique triangle so that side a is opposite angle oz 9 is opposite angle 6 and side 0 is opposite angle 7 3 Solve an oblique triangle To nd the lengths of its sides and the measurement of its angleswe need the length of one side along with 1 two angles 2 one angle and one other side 3 the other two sides There are four possibilities Case 1 One side and two angles are known ASA or SAA Case 2 Two sides and the angle opposite one of them are known SSA Case 3 Two sides and the included angle are known SAS Case 4 Three sides are known SSS 4 Law of sines For a triangle with sides a b c and opposite angles oz 6 7 respectively Remark The law of sines is used to solve triangles for which Case 1 or Case 2 holds 5 Example Solve the triangle oz 4006 600a 4 b m 539 c m 613 6 Example Solve the triangle oz 3506 1506 5 a m 374 b z 169 7 The ambiguous case 1 N0 triangle if a lt h bsina 2 One right triangle if a h bsina 3 Two triangle if a lt b and h bsina lt a 4 One triangle if a 2 b 8 Example Solve the triangle a 31 2 oz 400 c m 424 9 Example Solve the triangle a 61 8a 350 01 m 1042 c2 269 10 Example Solve the triangle a 2 c 1 7 500 43 The laws of cosines 1 Case 3 Two sides and the included angle are known SAS Case 4 Three side are known SSS 2 Law of cosines For a triangle with sides a b c and oppo site angles oz respectively 3 Remark 1 Law of cosines The square of one side of a triangle equal the sum of the squares of the other two side minus twice their product times the cosine of their included angle 2 Special case Pythagorean Theorem 4 Example Solve the triangle a 21 37 600 oz 4090 b 7910 5 Example Solve the triangle a 3630 6 m 2640 7 w 11730 44 Area of a triangle 1 The area A of a triangle is where is b is the base and h is an altitude drawn to that base 2 Other formulas 3 Remark The area A of a triangle equals onehalf the product of two of its sides times the sine of their included angle 4 Example Find the area A of the triangle for which a8b6 and7300 5 Heron s Formula The area A of a triangle with sides a b and c is 9 where s a b c 6 Example Find the area of a triangle whose sides are 4 5 and 6
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