Chapter 5 Analytic Trigonometry Copyright © 2014, 2010, 2007 Pearson Education, Inc Verifying Trigonometric Identities

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Objectives: Use the fundamental trigonometric identities to simplify trigonometric expressions. Use the fundamental trigonometric identities to verify identities.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 The Fundamental Identities

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Simplifying Trigonometric Expressions

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Simplifying Trigonometric Expressions Identities enable us to write the same expression in different ways. It is often possible to rewrite a complicated-looking expression as a much simpler one. To simplify algebraic expressions, we used factoring, common denominators, and the Special Product Formulas. To simplify trigonometric expressions, we use these same techniques together with the fundamental trigonometric identities.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example 1 – Simplifying a Trigonometric Expression Simplify the expression cos t + tan t sin t. Solution: We start by rewriting the expression in terms of sine and cosine: cos t + tan t sin t = cos t + sin t = = = sec t Quotient identity Common denominator Pythagorean identity Reciprocal identity

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Verifying an Identity

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Using Fundamental Identities to Verify Other Identities To verify an identity, we show that one side of the identity can be simplified so that it is identical to the other side. Each side of the equation is manipulated independently of the other side of the equation. Start with the side containing the more complicated expression. If you substitute one or more of the fundamental identities on the more complicated side, you will often be able to rewrite it in a form identical to that of the other side.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Changing to Sines and Cosines to Verify an Identity Verify the identity: Divide the numerator and the denominator by the common factor. Multiply the remaining factors in the numerator and denominator. The identity is verified.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Using Factoring to Verify an Identity Verify the identity: Factor sin x from the two terms. Multiply. The identity is verified.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Combining Fractional Expressions to Verify an Identity Verify the identity: The least common denominator is sin x(1 + cos x) Use FOIL to multiply (1 + cos x)(1 + cos x)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Combining Fractional Expressions to Verify an Identity (continued) Add the numerators. Put this sum over the LCD. Regroup terms in the numerator. Add constant terms in the numerator. Verify the identity:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Combining Fractional Expressions to Verify an Identity (continued) Verify the identity: Factor and simplify. Factor out the constant term. The identity is verified.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Working with Both Sides Separately to Verify an Identity Verify the identity: We begin by working with the left side Rewrite each side with the LCD. Add numerators. Put this sum over the LCD. Simplify the numerator. Multiply the factors in the denominator.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Working with Both Sides Separately to Verify an Identity (continued) Verify the identity: We work with the right side. Rewrite each numerator with the LCD. Add numerators. Put this sum over the LCD.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Working with Both Sides Separately to Verify an Identity (continued) Verify the identity: We are working with the right side of the identity. Factor out the constant term, 2.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Working with Both Sides Separately to Verify an Identity (continued) Verify the identity: Working with the left side, we found that Working with the right side, we found that The identity is verified because both sides are equal.