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MATHPOWER TM 12, WESTERN EDITION

Chapter 5 Trigonometric Equations. 5.4. Trigonometric Identities. 5.4. 1. MATHPOWER TM 12, WESTERN EDITION. Trigonometric Identities. A trigonometric equation is an equation that involves at least one trigonometric function of a variable. The

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MATHPOWER TM 12, WESTERN EDITION

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  1. Chapter 5 Trigonometric Equations 5.4 Trigonometric Identities 5.4.1 MATHPOWERTM 12, WESTERN EDITION

  2. Trigonometric Identities A trigonometric equation is an equation that involves at least one trigonometric function of a variable. The equation is a trigonometric identity if it is true for all values of the variable for which both sides of the equation are defined. Prove that Recall the basic trig identities: 5.4.2

  3. Trigonometric Identities Quotient Identities Reciprocal Identities Pythagorean Identities 5.4.3

  4. Trigonometric Identities [cont’d] sinx x sinx = 5.4.4

  5. Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. b) a) 5.4.5

  6. Simplifying Trigonometric Expressions c) (1 + tan x)2 - 2 sin x sec x d) 5.4.6

  7. Proving an Identity Steps in Proving Identities 1. Start with the more complex side of the identity and work with it exclusively to transform the expression into the simpler side of the identity. 2. Look for algebraic simplifications: 3. Look for trigonometric simplifications: 4. Keep the simpler side of the identity in mind. 5.4.7

  8. Proving an Identity Prove the following: a) sec x(1 + cos x) = 1 + sec x 1 + sec x c) tan x sin x + cos x = sec x b) sec x = tan x csc x 5.4.8

  9. Proving an Identity d) sin4x - cos4x = 1 - 2cos2x 1 - 2cos2x e) 5.4.9

  10. Proving an Identity f) 5.4.10

  11. Using Exact Values to Prove an Identity Consider a) Use a graph to verify that the equation is an identity. b) Verify that this statement is true for x = c) Use an algebraic approach to prove that the identity is true in general. State any restrictions. a) 5.4.11

  12. Using Exact Values to Prove an Identity [cont’d] b) Verify that this statement is true for x = Rationalize the denominator: 5.4.12

  13. Using Exact Values to Prove an Identity [cont’d] c) Use an algebraic approach to prove that the identity is true in general. State any restrictions. Restrictions: Note the left side of the equation has the restriction Therefore, where n is any integer. The right side of the equation has the restriction Therefore, And , where n is any integer. 5.4.13

  14. Proving an Equation is an Identity Consider the equation a) Use a graph to verify that the equation is an identity. b) Verify that this statement is true for x = 2.4 rad. c) Use an algebraic approach to prove that the identity is true in general. State any restrictions. a) 5.4.14

  15. Proving an Equation is an Identity [cont’d] b) Verify that this statement is true for x = 2.4 rad. 5.4.15

  16. Proving an Equation is an Identity [cont’d] c) Use an algebraic approach to prove that the identity is true in general. State any restrictions. Note the left side of the equation has the restriction: sin2A - sin A ≠ 0 sin A(sin A - 1) ≠ 0 The right side of the equation has the restriction sin A ≠ 0, or A ≠ 0. Therefore, A ≠ 0, p + 2pn, where n is any integer. L.S. = R.S. 5.4.16

  17. Assignment Suggested Questions: Pages 264 and 265 A 1-10, 21-25, 37, 11, 13, 16 B 12, 20, 26-34 5.4.16

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