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Chapter 5 Trigonometric Equations 5.4 MATHPOWERTM 12, WESTERN EDITION 5.4.1 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: y sin r x cos r y tan x sin Prove that tan . cos y y x x r r y r r x y x L.S. = R.S. 5.4.2 Trigonometric Identities Quotient Identities cos cot sin sin tan cos Reciprocal Identities 1 sin csc 1 cos sec 1 tan cot Pythagorean Identities sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2 sin2 = 1 - cos2 cos2 = 1 - sin2 tan2 = sec2 - 1 cot2 = csc2 - 1 5.4.3 Trigonometric Identities [cont’d] sinx x sinx = sin2x 1 cos 2 1 cos 2 1 cos cos cos cos cos sin A cos A sin2 A 2sin Acos A cos 2 A 2 1 2sin Acos A cos A sin A cos A sin A 1 sin A 1 sin A = cosA 5.4.4 Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. a) cos sin tan sin cos sin cos sin2 cos cos cos sin cos 1 cos 2 sec 2 cot 2 1 sin2 b) cos 2 2 sin cos 2 1 cos 2 1 2 sin cos2 1 2 sin csc 2 5.4.5 Simplifying Trigonometric Expressions c) (1 + tan x)2 - 2 sin x sec x 1 cos x sin x 2 1 2 tan x tan x 2 cos x (1 tan x) 2 sin x 2 1 tan2 x 2tanx 2 tanx sec2 x csc x d) tan x cot x 1 sin x sin x cos x cos x sin x 1 sin x sin 2 x cos 2 x sin xcos x 1 sin x 1 sin x cos x 1 sin x cos x sin x 1 cos x 5.4.6 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 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 secx sin x 1 cos x sin x 1 cos x secx L.S. = R.S. c) tan x sin x + cos x = sec x secx sin x sin x cos x cos x 1 sin2 x cos 2 x cos x 1 cos x L.S. = R.S. secx 5.4.8 Proving an Identity sin4x - cos4x d) = 1 - 2cos2 x = (sin2x - cos2x)(sin2x + cos2x) 1 - 2cos2x = (1 - cos2x - cos2x) = 1 - 2cos2x L.S. = R.S. 1 1 2 e) 2 cs c x 1 cos x 1 cos x (1 cos x) (1 cos x) 2 csc2 x (1 cos x)(1 cos x) 2 2 (1 cos x) 2 2 sin x 2 csc2 x L.S. = R.S. 5.4.9 Proving an Identity f) cos A 1 sin A 1 sin A cos A cos 2 A (1 sinA)(1 sin A) (1 sin A)(cos A) 2 secA 2 secA cos 2 A (1 2sin A sin2 A) (1 sinA)(cos A) cos 2 A sin2 A 1 2sin A (1 sinA)(cos A) 2 2sin A (1 sin A)(cos A) 2(1 sin A) (1 sin A)(cos A) 2 (cos A) 2sec A L.S. = R.S. 5.4.10 Using Exact Values to Prove an Identity Consider sin x 1 cos x . 1 cos x sin x a) Use a graph to verify that the equation is an identity. b) Verify that this statement is true for x = . 6 c) Use an algebraic approach to prove that the identity is true in general. State any restrictions. a) 1 sin cos xx y 1 sin cos xx 5.4.11 Using Exact Values to Prove an Identity [cont’d] b) Verify that this statement is true for x = . sin x 1 cos x sin 1 cos x sin x 6 1 cos 6 1 2 3 1 2 1 2 2 2 3 1 2 3 2 3 1 cos sin 1 6 Rationalize the denominator: 1 2 3 6 6 3 2 1 2 2 3 2 2 1 2 3 L.S. = R.S. 1 2 3 2 3 2 3 2 3 43 2 3 Therefore, the identity is true for the particular x . case of 6 5.4.12 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: sin x 1 cos x Note the left side of the 1 cos x sin x equation has the restriction sin x 1 cos x 1 cos x 1 - cos x ≠ 0 or cos x ≠ 1. Therefore, x ≠ 0 + 2 n, 1 cos x 1 cos x sin x where n is any integer. sin x(1 cos x) 1 cos x 2 The right side of the equation has the restriction sin x ≠ 0. x = 0 and Therefore, x ≠ 0 + 2n and x ≠ + 2n, where n is any integer. sin x(1 cos x) 2 sin x 1 cos x sin x L.S. = R.S. 5.4.13 Proving an Equation is an Identity 2 sin A 1 1 Consider the equation 1 . 2 sin A sin A sin A 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) sin21A 1 y 1 2 sin sin A Asin A 5.4.14 Proving an Equation is an Identity [cont’d] b) Verify that this statement is true for x = 2.4 rad. sin2 A 1 sin2 A sin A 1 1 sin A (sin 2.4)2 1 2 (sin 2.4) sin2.4 1 1 sin 2.4 = 2.480 466 = 2.480 466 L.S. = R.S. Therefore, the equation is true for x = 2.4 rad. 5.4.15 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 sin2 A 1 sin2 A sin A 1 1 sin A (sin A 1)(sin A 1) sin A(sin A 1) 1 1 sin A (sin A 1) sin A sin A 1 sin A sin A 1 1 sin A L.S. = R.S. equation has the restriction: sin2A - sin A ≠ 0 sin A(sin A - 1) ≠ 0 sin A ≠ 0 or sin A ≠ 1 A 0, or A 2 Therefore,A 0 2 n or A + 2 n, or A 2 n, wheren is 2 any integer. The right side of the equation has the restriction sin A ≠ 0, or A ≠ 0. Therefore, A ≠ 0, + 2 n, where n is any integer. 5.4.16 Suggested Questions: Pages 264 and 265 A 1-10, 21-25, 37, 11, 13, 16 B 12, 20, 26-34 5.4.16