Multiple-Angle and Product-to-Sum Formulas

Transcription

1 Multiple-Angle and Product-to-Sum Formulas MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 011

2 Objectives In this lesson we will learn to: use multiple-angle formulas to rewrite and evaluate trigonometric functions, use power-reducing formulas to rewrite and evaluate trigonometric functions, use half-angle formulas to rewrite and evaluate trigonometric functions, use product-to-sum and sum-to-product formulas to rewrite and evaluate trigonometric functions, use trigonometric formulas to rewrite real-life models.

3 Background Recall the sum and difference of angle formulas we have learned previously: sin(u + v) = sin u cos v + cos u sin v sin(u v) = sin u cos v cos u sin v cos(u + v) = cos u cos v sin u sin v cos(u v) = cos u cos v + sin u sin v tan(u + v) = tan u + tan v 1 tan u tan v tan(u v) = tan u tan v 1 + tan u tan v

4 Double-Angle Formulas A double-angle is generally expressed as u. Thus the double-angle formulas are: sin u = sin u cos u

5 Double-Angle Formulas A double-angle is generally expressed as u. Thus the double-angle formulas are: sin u = sin u cos u cos u = cos u sin u tan u = tan u 1 tan u

6 Find the solutions to the following equation in the interval [0, π). sin x + cos x = 0

7 Find the solutions to the following equation in the interval [0, π). sin x + cos x = 0 sin x cos x + cos x = 0 cos x( sin x + 1) = 0 cos x = 0 or sin x = 1 x = π, 3π, 7π 6, 11π 6

8 Find the exact values of sin u, cos u, and tan u given that cot u = 5 and 3π/ < u < π.

9 Find the exact values of sin u, cos u, and tan u given that cot u = 5 and 3π/ < u < π. Using right triangle trigonometry we see that sin u = 6 6, cos u = 5 6 6, tan u = 1 5.

10 Find the exact values of sin u, cos u, and tan u given that cot u = 5 and 3π/ < u < π. Using right triangle trigonometry we see that sin u = 6 6, cos u = 5 6 6, tan u = 1 5. ( ) ( 6 5 ) 6 sin u = sin u cos u = = cos u = cos u sin u = = 1 13 tan u tan u = 1 tan u = /5 1 1/5 = 5 1

11 Verify the following identity. (sin x + cos x) = 1 + sin x

12 Verify the following identity. (sin x + cos x) = 1 + sin x sin x + sin x cos x + cos x = (sin x + cos x) + ( sin x cos x) = 1 + sin x =

13 Power-Reducing Formulas Power-reducing formulas replace powers of trigonometric functions with simpler expressions. Power-Reducing Formulas sin u = cos u = tan u = 1 cos u 1 + cos u 1 cos u 1 + cos u

14 Use power-reducing formulas to rewrite the following expression in terms of the first power of cosine. sin x cos x =

15 Use power-reducing formulas to rewrite the following expression in terms of the first power of cosine. ( ) ( ) 1 cos x 1 + cos x sin x cos x = = 1 (1 cos x)(1 + cos x) 4 = 1 4 (1 cos x) = 1 4 sin x ( ) 1 cos 4x = 1 4 = 1 (1 cos 4x) 8

16 Half-Angle Formulas Closely related to the power-reducing formulas are the half-angle formulas which are obtained by replacing u by u/. Half-Angle Formulas sin u = ± 1 cos u cos u = ± 1 + cos u tan u = 1 cos u sin u = sin u 1 + cos u The signs of sin u and cos u u falls. depend on the quadrant in which

17 Use the half-angle formulas to determine the exact values of sin 165 = cos 165 = tan 165 =

18 Use the half-angle formulas to determine the exact values of ( ) cos 330 sin / = sin = = cos 165 = tan 165 =

19 Use the half-angle formulas to determine the exact values of ( ) cos 330 sin / = sin = = ( ) cos 330 cos / = cos = = tan 165 =

20 Use the half-angle formulas to determine the exact values of ( ) cos 330 sin / = sin = = ( ) cos 330 cos / = cos = = tan / = 1 + 3/ = 3 + 3

21 Product-to-Sum Formulas If we combine the sum and difference of angles formulas we can derive the product-to-sum formulas. Product-to-Sum Formulas sin u sin v = 1 [cos(u v) cos(u + v)] cos u cos v = 1 [cos(u v) + cos(u + v)] sin u cos v = 1 [sin(u + v) + sin(u v)] cos u sin v = 1 [sin(u + v) sin(u v)]

22 Use the product-to-sum formulas to write the following as a sum or difference. sin π 4 cos π 1 =

23 Use the product-to-sum formulas to write the following as a sum or difference. sin π 4 cos π 1 = 1 [ ( π sin 4 1) + π ( π + sin 4 1)] π = 1 [ ( π ) ( π )] sin + sin 3 6

24 Use the product-to-sum formulas to write the following as a sum or difference. sin π 4 cos π 1 = 1 [ ( π sin 4 1) + π ( π + sin 4 1)] π = 1 [ ( π ) ( π )] sin + sin 3 6 [ ] = = 4

25 Sum-to-Product Formulas Occasionally we may wish to rewrite a sum or difference of trigonometric functions as a product. Sum-to-Product Formulas ( ) ( ) u + v u v sin u + sin v = sin cos ( ) ( ) u + v u v sin u sin v = cos cos ( ) ( ) u + v u v cos u + cos v = cos cos ( ) ( ) u + v u v cos u cos v = sin sin

26 Use the sum-to-product formulas to find the exact value of the following expression. cos 10 + cos 60 =

27 Use the sum-to-product formulas to find the exact value of the following expression. cos 10 + cos 60 = ( ) ( ) cos cos = cos (90 ) cos (30 ) = 0

28 Application The range of a projectile fired at an angle θ with respect to the horizontal and with an initial velocity of v 0 feet per second is r = 1 3 v 0 sin θ. If an athlete throws a javelin at 75 feet per second, at what angle must the javelin be thrown so that it travels 130 feet?

29 Solution 130 = 1 3 (75) sin θ 4160 = 565 sin θ = sin θ θ = arcsin( ) = θ 3.85

30 Homework Read Section 5.5. Exercises: 1, 5, 9, 13,..., 133, 137