1. Copyright © 2009 Pearson Addison-Wesley 5.2-1 5 Trigonometric Identities

2. Copyright © 2009 Pearson Addison-Wesley 5.2-2 5.1 Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Sum and Difference Identities for Cosine 5.4 Sum and Difference Identities for Sine and Tangent 5.5Double-Angle Identities 5.6Half-Angle Identities 5 Trigonometric Identities

3. Copyright © 2009 Pearson Addison-Wesley1.1-3 5.2-3 Verifying Trigonometric Identities 5.2 Verifying Identities by Working With One Side ▪ Verifying Identities by Working With Both Sides

4. Copyright © 2009 Pearson Addison-Wesley1.1-4 5.2-4 Hints for Verifying Identities  Learn the fundamental identities. 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.  Try to rewrite the more complicated side of the equation so that it is identical to the simpler side.

5. Copyright © 2009 Pearson Addison-Wesley1.1-5 5.2-5 Hints for Verifying Identities  It is sometimes helpful to express all trigonometric functions in the equation in terms of sine and cosine and then simplify the result.  Usually, any factoring or indicated algebraic operations should be performed. For example, the expression can be factored as

6. Copyright © 2009 Pearson Addison-Wesley1.1-6 5.2-6 Hints for Verifying Identities The sum or difference of two trigonometric expressions such as can be added or subtracted in the same way as any other rational expression.

7. Copyright © 2009 Pearson Addison-Wesley1.1-7 5.2-7 Hints for Verifying Identities  As you select substitutions, keep in mind the side you are not changing, because it represents your goal. For example, to verify the identity find an identity that relates tan x to cos x. Since and the secant function is the best link between the two sides.

8. Copyright © 2009 Pearson Addison-Wesley1.1-8 5.2-8 Hints for Verifying Identities  If an expression contains 1 + sin x, multiplying both the numerator and denominator by 1 – sin x would give 1 – sin2 x, which could be replaced with cos2x. Similar results for 1 – sin x, 1 + cos x, and 1 – cos x may be useful.

9. Copyright © 2009 Pearson Addison-Wesley1.1-9 5.2-9 Caution Verifying identities is not the same a solving equations. Techniques used in solving equations, such as adding the same terms to both sides, should not be used when working with identities since you are starting with a statement that may not be true.

10. Copyright © 2009 Pearson Addison-Wesley 5.2-10 Verifying Identities by Working with One Side To avoid the temptation to use algebraic properties of equations to verify identities, one strategy is to work with only one side and rewrite it to match the other side.

11. Copyright © 2009 Pearson Addison-Wesley1.1-11 5.2-11 Example 1 VERIFYING AN IDENTITY (WORKING WITH ONE SIDE) Verify that is an identity. Work with the right side since it is more complicated. Right side of given equation Distributive property Left side of given equation

12. Copyright © 2009 Pearson Addison-Wesley1.1-12 5.2-12 Example 2 VERIFYING AN IDENTITY (WORKING WITH ONE SIDE) Verify that is an identity. Distributive property Left side Right side

13. Copyright © 2009 Pearson Addison-Wesley1.1-13 5.2-13 Example 3 VERIFYING AN IDENTITY (WORKING WITH ONE SIDE) Verify that is an identity.

14. Copyright © 2009 Pearson Addison-Wesley1.1-14 5.2-14 Example 4 VERIFYING AN IDENTITY (WORKING WITH ONE SIDE) Verify that is an identity. Multiply by 1 in the form

15. Copyright © 2009 Pearson Addison-Wesley 5.2-15 Verifying Identities by Working with Both Sides If both sides of an identity appear to be equally complex, the identity can be verified by working independently on each side until they are changed into a common third result. Each step, on each side, must be reversible.

16. Copyright © 2009 Pearson Addison-Wesley1.1-16 5.2-16 Example 5 VERIFYING AN IDENTITY (WORKING WITH BOTH SIDES) Verify that is an identity. Working with the left side: Multiply by 1 in the form Distributive property

17. Copyright © 2009 Pearson Addison-Wesley1.1-17 5.2-17 Example 5 VERIFYING AN IDENTITY (WORKING WITH BOTH SIDES) (continued) Working with the right side: Factor the numerator. Factor the denominator.

18. Copyright © 2009 Pearson Addison-Wesley1.1-18 5.2-18 Example 5 VERIFYING AN IDENTITY (WORKING WITH BOTH SIDES) (continued) So, the identity is verified. Right side of given equation Left side of given equation Common third expression

19. Copyright © 2009 Pearson Addison-Wesley1.1-19 5.2-19 Example 6 APPLYING A PYTHAGOREAN IDENTITY TO RADIOS Tuners in radios select a radio station by adjusting the frequency. A tuner may contain an inductor L and a capacitor. The energy stored in the inductor at time t is given by and the energy in the capacitor is given by where f is the frequency of the radio station and k is a constant.

20. Copyright © 2009 Pearson Addison-Wesley1.1-20 5.2-20 Example 6 APPLYING A PYTHAGOREAN IDENTITY TO RADIOS (continued) The total energy in the circuit is given by Show that E is a constant function.* *(Source: Weidner, R. and R. Sells, Elementary Classical Physics, Vol. 2, Allyn & Bacon, 1973.)

21. Copyright © 2009 Pearson Addison-Wesley1.1-21 5.2-21 Example 6 APPLYING A PYTHAGOREAN IDENTITY TO RADIOS (continued) Factor.