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You found trigonometric values using the unit circle. (Lesson 4-3) • LEQ: How do we identify and use basic trigonometric identities to find trigonometric values & use basic trigonometric identities to simplify and rewrite trigonometric expressions? • identity • trigonometric identity • cofunction • odd-even identities Use Reciprocal and Quotient Identities A. If , find sec θ. Reciprocal Identity Divide. Answer: Use Reciprocal and Quotient Identities A. If , find sec θ. Reciprocal Identity Divide. Answer: Use Reciprocal and Quotient Identities B. If and , find sin x. Reciprocal Identity Quotient Identity Substitute for cos x. Use Reciprocal and Quotient Identities Divide. Multiply each side by Simplify. Answer: . Use Reciprocal and Quotient Identities Divide. Multiply each side by Simplify. Answer: . If A. B. C. D. , find sin . If A. B. C. D. , find sin . Use Pythagorean Identities If cot θ = 2 and cos θ < 0, find sin θ and cos θ. Use the Pythagorean Identity that involves cot θ. cot 2 θ + 1 = csc 2 θ Pythagorean Identity (2) 2 + 1 = csc 2 θ cot θ = 2 5 = csc 2 θ Simplify. = csc θ Take the square root of each side. Reciprocal Identity Solve for sin θ. Use Pythagorean Identities Since is positive and cos θ < 0, sin θ must be negative. So . You can then use this quotient identity again to find cos θ. Quotient Identity cot θ = 2 and Multiply each side by . Use Pythagorean Identities So, Answer: Use Pythagorean Identities So, Answer: Check sin 2 θ + cos 2 θ = 1 Pythagorean Identity Simplify. Find the value of csc and cot if cos < 0. A. B. C. D. and Find the value of csc and cot if cos < 0. A. B. C. D. and Use Cofunction and Odd-Even Identities If cos x = –0.75, find Factor. Odd-Even Identity Cofunction Identity cos x = –0.75 Simplify. Use Cofunction and Odd-Even Identities So, Answer: = 0.75. Use Cofunction and Odd-Even Identities So, = 0.75. Answer: 0.75 If cos x = 0.73, find A. –0.73 B. –0.68 C. 0.68 D. 0.73 . If cos x = 0.73, find A. –0.73 B. –0.68 C. 0.68 D. 0.73 . Simplify by Rewriting Using Only Sine and Cosine Simplify . Solve Algebraically Pythagorean Identity Multiply. Simplify. So, = cos x. Simplify by Rewriting Using Only Sine and Cosine Support Graphically The graphs of appear to be identical. Answer: and y = cos x Simplify by Rewriting Using Only Sine and Cosine Support Graphically The graphs of appear to be identical. Answer: cos x and y = cos x Simplify csc x – cos x cot x. A. cot x B. tan x C. cos x D. sin x Simplify csc x – cos x cot x. A. cot x B. tan x C. cos x D. sin x Simplify by Factoring Simplify cos x tan x – sin x cos 2 x. Solve Algebraically cos x tan x – sin x cos 2 x Original expression Quotient Identity Multiply. Factor. Pythagorean Identity = sin3x Simplify. So, cos x tan x – sin x cos 2 x = sin3 x. Simplify by Factoring Support Graphically The graphs below appear to be identical. Answer: Simplify by Factoring Support Graphically The graphs below appear to be identical. Answer: sin3 x Simplify cos2 x sin x – cos(90° – x). A. –sin3 x B. sin3 x C. cos2 x – 1 D. sin x cos x Simplify cos2 x sin x – cos(90° – x). A. –sin3 x B. sin3 x C. cos2 x – 1 D. sin x cos x Simplify by Combining Fractions Simplify . Common denominator Multiply. Add the numerators. Simplify. Pythagorean Identity Simplify by Combining Fractions Reciprocal Identity Reciprocal and Quotient Identities Divide out common factor. Reciprocal Identity –2csc2 x. Answer: Simplify by Combining Fractions Reciprocal Identity Reciprocal and Quotient Identities Divide out common factor. Reciprocal Identity –2csc2 x. Answer: – 2 sec2 x – 2 sec2 x Simplify A. cos x B. 2 + 2 cos x C. 2 sin x D. 2 csc x . Simplify A. cos x B. 2 + 2 cos x C. 2 sin x D. 2 csc x . Rewrite to Eliminate Fractions Rewrite as an expression that does not involve a fraction. Pythagorean Identity Reciprocal Identity Reciprocal Identity Quotient Identity Rewrite to Eliminate Fractions So, Answer: = tan2 x. Rewrite to Eliminate Fractions So, = tan2 x. Answer: tan 2 x Rewrite involve a fraction. A. –2 tan2 x B. 1+ sin x C. 1 – cos x D. 2 sin x as an expression that does not Rewrite involve a fraction. A. –2 tan2 x B. 1+ sin x C. 1 – cos x D. 2 sin x as an expression that does not