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Unit 34 TRIGONOMETRIC FUNCTIONS WITH RIGHT TRIANGLES
2 VARIATION OF FUNCTIONS As the size of an angle increases, the sine, tangent, and secant functions increase, but the cofunctions (cosine, cotangent, cosecant) decrease Which is greater: cos 38° or cos 43°? Since the cosine function decreases as the size of the angle increases, cos 38° is greater than cos 43° Which is greater: tan 42° or tan 24°? Since the tangent function increases as the size of the angle increases, tan 42° is greater than tan 24°
3 FUNCTIONS OF COMPLEMENTARY ANGLES Two angles are complementary when their sum is 90°. For example, 30° is the complement of 60°, and 60° is the complement of 30° A function of an angle is equal to the cofunction of the complement of the angle csc A = sec (90° – A)sec A = csc (90° – A) cot A = tan (90° – A)tan A = cot (90° – A) cos A = sin (90° – A)sin A = cos (90° – A)
4 FINDING UNKNOWN ANGLES Procedure for determining an unknown angle when two sides are given: –In relation to the desired angle, identify two given sides as adjacent, opposite, or hypotenuse –Determine the functions that are ratios of the sides identified in relation to the desired angle –Choose one of the two functions, substitute the given sides in the ratio –Determine the angle that corresponds to the quotient of the ratio
5 EXAMPLE OF FINDING AN ANGLE Determine B to the nearest tenth of a degree in the triangle below: B 5.7" 3.2"
6 EXAMPLE (Cont) –Since 5.7" is opposite B and 3.2" is adjacent B; either the tangent or the cotangent function could be used –Remember that when looking for an angle you will use arctan in this case or tan -1 on your calculator –Choosing the tangent function:
7 FINDING UNKNOWN SIDES Procedure for determining an unknown side when an angle and a side are given: –In relation to the given angle, identify the given side and the unknown side as adjacent, opposite, or hypotenuse –Determine the trigonometric functions that are ratios of the sides identified in relation to the given angle –Choose one of the two functions and substitute the given side and given angle –Solve as a proportion for the unknown side
8 EXAMPLE OF FINDING A SIDE Determine side x (to the nearest hundredth) of the right triangle shown below: 46.3° x 2.7 cm
9 EXAMPLE (Cont) –In relation to the 46.3° angle, the 2.7 cm side is the adjacent side and side x is the hypotenuse. Thus, either the cosine or the secant function could be used –Choosing the cosine function: so x = 3.91 cm Ans
10 PRACTICE PROBLEMS 1. Which is greater: sin 48° or sin 32°? 2. Which is greater: csc 54.3° or csc 45.3°? 3. What is the cofunction of the complement of sec 35°? 4. What is the cofunction of the complement of cos 82°?
11 PRACTICE PROBLEMS (Cont) 5. Determine angle A to the nearest tenth of a degree in the triangle shown below: A 4.4" 3.2"
12 PRACTICE PROBLEMS (cont) 6. Determine side b in the triangle given below. Round your answer to two decimal places. 7. Determine 1 (to the nearest tenth of a degree) in the triangle given below. 11 3.25" 5.86" b 5.4 mm 28°
13 PROBLEM ANSWER KEY 1. sin 48° 2. csc 45.3° 3. csc 55° 4. sin 8° 5. 46.7° 6. 4.77 mm 7. 33.7°
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