Copyright © 2005 Pearson Education, Inc.
Chapter 5 Trigonometric Identities
Copyright © 2005 Pearson Education, Inc. 5.1 Fundamental Identities
Copyright © 2005 Pearson Education, Inc. Slide 5-4 Fundamental Identities Reciprocal Identities Quotient Identities
Copyright © 2005 Pearson Education, Inc. Slide 5-5 More Identities Pythagorean Identities Negative-Angle Identities
Copyright © 2005 Pearson Education, Inc. Slide 5-6 Example: If and is in quadrant II, find each function value.
Copyright © 2005 Pearson Education, Inc. Slide 5-7 Example: Express One Function in Terms of Another Express cot x in terms of sin x.
Copyright © 2005 Pearson Education, Inc. Slide 5-8 Example: Rewriting an Expression in Terms of Sine and Cosine Rewrite cot tan in terms of sin and cos .
Copyright © 2005 Pearson Education, Inc. Slide
Copyright © 2005 Pearson Education, Inc. 5.2 Verifying Trigonometric Identities
Copyright © 2005 Pearson Education, Inc. Slide , 16, 20
Copyright © 2005 Pearson Education, Inc. Slide 5-12 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.
Copyright © 2005 Pearson Education, Inc. Slide 5-13 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 then simplify the result. 4. Usually, any factoring or indicated algebraic operations should be performed. For example, the expression can be factored as The sum or difference of two trigonometric expressions such as can be added or subtracted in the same way as any other rational expression.
Copyright © 2005 Pearson Education, Inc. Slide 5-14 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.
Copyright © 2005 Pearson Education, Inc. Slide 5-15 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 sin 2 x, which could be replaced with cos 2 x. 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.
Copyright © 2005 Pearson Education, Inc. Slide 5-16 Example: Working with One Side Prove the identity
Copyright © 2005 Pearson Education, Inc. Slide 5-17 Example: Working with One Side Prove the identity
Copyright © 2005 Pearson Education, Inc. Slide 5-18 Example: Working with One Side Prove the identity
Copyright © 2005 Pearson Education, Inc. Slide 5-19 Example: Working with Both Sides Verify that the following equation is an identity.
Copyright © 2005 Pearson Education, Inc. 5.3 Sum and Difference Identities for Cosine
Copyright © 2005 Pearson Education, Inc. Slide 5-21 Cosine of a Sum or Difference Find the exact value of cos 75 .
Copyright © 2005 Pearson Education, Inc. Slide 5-22 More Examples
Copyright © 2005 Pearson Education, Inc. Slide 5-23 Cofunction Identities Similar identities can be obtained for a real number domain by replacing 90 with /2.
Copyright © 2005 Pearson Education, Inc. Slide 5-24 Example: Using Cofunction Identities Find an angle that satisfies sin ( 20 ) = cos 38
Copyright © 2005 Pearson Education, Inc. Slide 5-25 Example: Reducing Write cos (270 ) as a trigonometric function of .
Copyright © 2005 Pearson Education, Inc. Slide
Copyright © 2005 Pearson Education, Inc. 5.4 Sum and Difference Identities for Sine and Tangent
Copyright © 2005 Pearson Education, Inc. Slide 5-28 Sine of a Sum of Difference Tangent of a Sum or Difference
Copyright © 2005 Pearson Education, Inc. Slide 5-29 Example: Finding Exact Values Find an exact value for sin 105 .
Copyright © 2005 Pearson Education, Inc. Slide 5-30 Example: Finding Exact Values continued Find an exact value for sin 90 cos 135 cos 90 sin 135
Copyright © 2005 Pearson Education, Inc. Slide 5-31 Example: Write each function as an expression involving functions of . sin (30 + ) tan (45 + )
Copyright © 2005 Pearson Education, Inc. Slide 5-32 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), tan (A + B), and the quadrant of A + B.
Copyright © 2005 Pearson Education, Inc. Slide
Copyright © 2005 Pearson Education, Inc. 5.5 Double-Angle Identities
Copyright © 2005 Pearson Education, Inc. Slide 5-35 Double-Angle Identities Try 6, 26
Copyright © 2005 Pearson Education, Inc. Slide 5-36 Example: Given tan = 3/5 and sin < 0, find sin 2 , and cos 2 . Problems like 7-15 Find the value of sin .
Copyright © 2005 Pearson Education, Inc. Slide 5-37 Example: Multiple-Angle Identity Find an equivalent expression for cos 4x.
Copyright © 2005 Pearson Education, Inc. Slide ,54
Copyright © 2005 Pearson Education, Inc. Slide 5-39 Product-to-Sum Identities
Copyright © 2005 Pearson Education, Inc. Slide 5-40 Example Write 3sin 2 cos as the sum or difference of two functions.
Copyright © 2005 Pearson Education, Inc. Slide 5-41 Sum-to-Product Identities
Copyright © 2005 Pearson Education, Inc. 5.6 Half-Angle Identities
Copyright © 2005 Pearson Education, Inc. Slide 5-43 Half-Angle Identities
Copyright © 2005 Pearson Education, Inc. Slide 5-44 Example: Finding an Exact Value Use a half-angle identity to find the exact value of sin ( /8).
Copyright © 2005 Pearson Education, Inc. Slide 5-45 Example: Finding an Exact Value Find the exact value of tan 22.5
Copyright © 2005 Pearson Education, Inc. Slide , 30, 40, 54