1 / 47

Chapter 5

Chapter 5. Trigonometric Identities. 5.1. Fundamental Identities. Fundamental Identities. Reciprocal Identities Quotient Identities. More Identities. Pythagorean Identities Negative-Angle Identities. a) sec  Look for an identity that relates tangent and secant.

veta
Download Presentation

Chapter 5

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 5 Trigonometric Identities

  2. 5.1 Fundamental Identities

  3. Fundamental Identities • Reciprocal Identities • Quotient Identities

  4. More Identities • Pythagorean Identities • Negative-Angle Identities

  5. a) sec  Look for an identity that relates tangent and secant. Example: If and  is in quadrant II, find each function value.

  6. b) sin  c) cot () Example: If and  is in quadrant II, find each function value continued

  7. Example: Express One Function in Terms of Another • Express cot x in terms of sin x.

  8. Example: Rewriting an Expression in Terms of Sine and Cosine • Rewrite cot   tan  in terms of sin  and cos  .

  9. 5.2 Verifying Trigonometric Identities

  10. Hints for Verifying Identities • 1. Learn the fundamental identities given in the last section. Whenever you see either side of a fundamental identity, the other side should come to mind. Also, be aware of equivalent forms of the fundamental identities. For example is an alternative form of the identity • 2. Try to rewrite the more complicated side of the equation so that it is identical to the simpler side.

  11. Hints for Verifying Identities continued • 3. It is sometimes helpful to express all trigonometric functions in the equation in terms of sine and cosine and them simplify the result. • 4. Usually, any factoring or indicated algebraic operations should be performed. For example, the expression can be factored as The sum of difference of two trigonometric expressions such as can be added or subtracted in the same way as any other rational expression.

  12. Hints for Verifying Identities continued • 5. As you select substitutions, keep in mind the side you are changing, because it represents your goal. For example, to verify the identity try to think of an identity that relates tan x to cos x. In this case, since and the secant function is the best link between the two sides.

  13. Hints for Verifying Identities continued • 6. If an expression contains 1 + sin x, multiplying both the numerator and denominator by 1  sin x would give 1  sin2x, which could be replaced with cos2x. Similar results for 1  sin x, 1 + cos x, and 1  cos x may be useful. • Remember that verifying identities is NOT the same as solving equations.

  14. Example: Working with One Side • Prove the identity • Solution: Start with the left side.

  15. Prove the identity Solution—start with the right side continued Example: Working with One Side • continued

  16. Example: Working with One Side • Prove the identity • Start with the left side.

  17. Example: Working with Both Sides • Verify that the following equation is an identity. • Solution: Since both sides appear complex, verify the identity by changing each side into a common third expression.

  18. Example: Working with Both Sides continued • Left side: Multiply numerator and denominator by

  19. Example: Working with Both Sides continued • Right Side: Begin by factoring. • We have shown that verifying that the given equation is an identity.

  20. 5.3 Sum and Difference Identities for Cosine

  21. Cosine of a Sum or Difference • Find the exact value of cos 15.

  22. More Examples

  23. Cofunction Identities • Similar identities can be obtained for a real number domain by replacing 90 with /2.

  24. Example: Using Cofunction Identities • Find an angle that satisfies sin (30) = cos 

  25. Example: Reducing • Write cos (180  ) as a trigonometric function of .

  26. 5.4 Sum and Difference Identities for Sine and Tangent

  27. Sine of a Sum of Difference • Tangent of a Sum or Difference

  28. Example: Finding Exact Values • Find an exact value for sin 105.

  29. Example: Finding Exact Values continued • Find an exact value for sin 90 cos 135 cos 90 sin 135

  30. sin (30 + ) tan (45 + ) Example: Write each function as an expression involving function of .

  31. Example: Finding Function Values and the Quadrant of A + B • Suppose that A and B are angles in standard position, with sin A = 4/5, /2 < A < , and cos b = 5/13,  < B < 3/2. Find sin (A + B).

  32. 5.5 Double-Angle Identities

  33. Double-Angle Identities

  34. Find the value of sin. Use the double-angle identity for sine, Example: Given cos  = 3/5 and sin  < 0, find sin 2, cos 2, and tan 2.

  35. Use any of the forms for cos to find cos 2. Find tan 2. Example: Given cos  = 3/5 and sin  < 0, find sin 2, cos 2, and tan 2 continued or

  36. Example: Multiple-Angle Identity • Find an equivalent expression for cos 3x. • Solution

  37. Product-to-Sum Identities

  38. Example • Write sin 2 cos as the sum or difference of two functions.

  39. Sum-to-Product Identities

  40. Example • Write cos 2  cos 4 as a product of two functions.

  41. 5.6 Half-Angle Identities

  42. Half-Angle Identities

  43. Example: Finding an Exact Value • Find the sin (/8) exactly.

  44. Example: Finding an Exact Value • Find the exact value of tan 22.5 using the identity • Sinc 22.5 = replace A with 45.

  45. Example: Finding an Exact Value continued • Multiply the numerator and denominator by 2, than rationalize the denominator.

  46. Simplify: Solution: Example: Simplifying Expressions

More Related