Chapter 5 Verifying Trigonometric Identities

1. Chapter 5Verifying Trigonometric Identities

2. Basic Trigonometric Identities

3. Text Example Verify the identity: sec x cot x= csc x. Solution The left side of the equation contains the more complicated expression. Thus, we work with the left side. Let us express this side of the identity in terms of sines and cosines. Perhaps this strategy will enable us to transform the left side into csc x, the expression on the right. Apply a reciprocal identity: sec x = 1/cosx and a quotient identity: cot x = cosx/sinx. Divide both the numerator and the denominator by cos x, the common factor.

4. Solution We start with the more complicated side, the left side. Factor out the greatest common factor, cos x, from each of the two terms. cos x- cos x sin2x= cos x(1 - sin2x) Factor cos x from the two terms. Use a variation of sin2 x + cos2 x = 1. Solving for cos2 x, we obtain cos2 x = 1 – sin2 x. = cos x ·cos2x = cos3x Multiply. Text Example Verify the identity: cosx - cosxsin2x = cos3x.. We worked with the left and arrived at the right side. Thus, the identity is verified.

5. Guidelines for Verifying Trigonometric Identities • Work with each side of the equation independently of the other side. Start with the more complicated side and transform it in a step-by-step fashion until it looks exactly like the other side. • Analyze the identity and look for opportunities to apply the fundamental identities. Rewriting the more complicated side of the equation in terms of sines and cosines is often helpful. • If sums or differences of fractions appear on one side, use the least common denominator and combine the fractions. • Don't be afraid to stop and start over again if you are not getting anywhere. Creative puzzle solvers know that strategies leading to dead ends often provide good problem-solving ideas.

6. Example • Verify the identity: csc(x) / cot (x) = sec (x) Solution:

7. Example • Verify the identity: Solution:

8. Example • Verify the following identity: Solution:

9. Example cont. Solution:

10. Sum and Difference Formulas

11. The Cosine of the Difference of Two Angles The cosine of the difference of two angles equals the cosine of the first angle times the cosine of the second angle plus the sine of the first angle times the sine of the second angle.

12. Solution We know exact values for trigonometric functions of 60° and 45°. Thus, we write 15° as 60° - 45° and use the difference formula for cosines. cos l5° = cos(60° - 45°) = cos 60° cos 45° + sin 60° sin 45° cos(-) = cos  cos + sin  sin  Text Example • Find the exact value of cos 15° Substitute exact values from memory or use special triangles. Multiply. Add.

13. cos(-) = cos  cos + sin  sin  Text Example Find the exact value of cos 80° cos 20° + sin 80° sin 20°. Solution The given expression is the right side of the formula for cos( - ) with = 80° and  = 20°. cos 80° cos 20° + sin 80° sin 20° = cos (80° - 20°) = cos 60° = 1/2

14. Example • Find the exact value of cos(180º-30º) Solution

15. Example • Verify the following identity: Solution

16. Sum and Difference Formulas for Cosines and Sines

17. Example • Find the exact value of sin(30º+45º) Solution

18. Sum and Difference Formulas for Tangents The tangent of the sum of two angles equals the tangent of the first angle plus the tangent of the second angle divided by 1 minus their product. The tangent of the difference of two angles equals the tangent of the first angle minus the tangent of the second angle divided by 1 plus their product.

19. Example • Find the exact value of tan(105º) Solution • tan(105º)=tan(60º+45º)

20. Example • Write the following expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. Solution

21. Double-Angle and Half-Angle Formulas

22. Double-Angle Identities

23. Three Forms of the Double-Angle Formula for cos2

24. Power-Reducing Formulas

25. Example • Write an equivalent expression for sin4x that does not contain powers of trigonometric functions greater than 1. Solution

26. Half-Angle Identities

27. Text Example Find the exact value of cos 112.5°. Solution Because 112.5° =225°/2, we use the half-angle formula for cos /2 with = 225°. What sign should we use when we apply the formula? Because 112.5° lies in quadrant II, where only the sine and cosecant are positive, cos 112.5° < 0. Thus, we use the - sign in the half-angle formula.

28. Half-Angle Formulas for:

29. Example • Verify the following identity: Solution

30. Product-to-Sum and Sum-to-Product Formulas

31. Product-to-Sum Formulas

32. Example • Express the following product as a sum or difference: Solution

33. sin  sin  = 1/2 [cos( - ) - cos( + )] sin  cos  = 1/2[sin( + ) + sin( - )] Text Example Express each of the following products as a sum or difference. a. sin 8x sin 3xb. sin 4x cos x Solution The product-to-sum formula that we are using is shown in each of the voice balloons. a. sin 8x sin 3x= 1/2[cos (8x- 3x) - cos(8x+ 3x)] = 1/2(cos 5x- cos 11x) b. sin 4x cos x= 1/2[sin (4x+x) + sin(4x-x)] = 1/2(sin 5x+ sin 3x)

34. Sum-to-Product Formulas

35. Example • Express the difference as a product: Solution

36. Example • Express the sum as a product: Solution

37. Example • Verify the following identity: Solution

38. Trigonometric Equations

39. Equations Involving a Single Trigonometric Function To solve an equation containing a single trigonometric function: • Isolate the function on one side of the equation. • Solve for the variable.

40. x Trigonometric Equations y y = cos x 1 y = 0.5 x –4  –2  2  4  –1 cos x = 0.5 has infinitely many solutions for –< x <  y y = cos x 1 0.5 2  cos x = 0.5 has two solutions for 0 < x < 2 –1

41. This is the given equation. 3 sin x- 2 = 5 sin x- 1 Subtract 5 sin x from both sides. 3 sin x- 5 sin x- 2 = 5 sin x- 5 sin x – 1 Simplify. -2 sin x- 2 =-1 Add 2 to both sides. -2 sin x= 1 Divide both sides by -2 and solve for sin x. sin x= -1/2 Text Example Solve the equation: 3 sin x- 2 = 5 sin x- 1. Solution The equation contains a single trigonometric function, sin x. Step 1Isolate the function on one side of the equation. We can solve for sin x by collecting all terms with sin x on the left side, and all the constant terms on the right side.

42. Solution The given equation is in quadratic form 2t2+t- 1 = 0 with t= cos x. Let us attempt to solve the equation using factoring. This is the given equation. 2 cos2x+ cos x- 1 = 0 Factor. Notice that 2t2 + t – 1 factors as (2t – 1)(2t + 1). (2 cos x- 1)(cos x+ 1) = 0 Set each factor equal to 0. 2 cos x- 1= 0 or cos x+ 1 = 0 Solve for cos x. 2 cos x= 1 cos x= -1 Text Example Solve the equation: 2 cos2 x+ cos x- 1 = 0, 0 £x< 2p. cos x= 1/2 x=px= 2pppx=p The solutions in the interval [0, 2p) are p/3, p, and 5p/3.

43. Example • Solve the following equation: Solution:

44. Example • Solve the equation on the interval [0,2) Solution:

45. Example • Solve the equation on the interval [0,2) Solution:

46. Example • Solve the equation on the interval [0,2) Solution: