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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. Recall the basic trig identities: 5.4.2

  3. Trigonometric Identities Quotient Identities Reciprocal Identities Pythagorean Identities sin2q + cos2q = 1 tan2q + 1 = sec2q cot2q + 1 = csc2q sin2q = 1 - cos2q tan2q = sec2q - 1 cot2q = csc2q - 1 cos2q = 1 - sin2q 5.4.3

  4. Trigonometric Identities [cont’d] sinx x sinx = sin2x = cosA 5.4.4

  5. 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: • Do any multiplying , factoring, or squaring which is • obvious in the expression. • Reduce two terms to one, either add two terms or • factor so that you may reduce. 3. Look for trigonometric simplifications: • Look for familiar trig relationships. • If the expression contains squared terms, think • of the Pythagorean Identities. • Transform each term to sine or cosine, if the • expression cannot be simplified easily using other ratios. 4. Keep the simpler side of the identity in mind. 5.4.7

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

  7. Proving an Identity d) sin4x - cos4x = 1 - 2cos2x 1 - 2cos2x = (sin2x - cos2x)(sin2x + cos2x) = (1 - cos2x - cos2x) = 1 - 2cos2x L.S. = R.S. e) L.S. = R.S. 5.4.9

  8. Proving an Identity f) L.S. = R.S. 5.4.10

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

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

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