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1 c Marcia Drost, March 10, 2008 150 Lecture Notes - Section 7.1 Trigonometric Identities 5 Given csc θ = , 3 tan θ = 3 4 Find the other four trigonometric functions. 1 Solution: csc u = sin u 1 1 3 sin u = = 5 = csc u 5 3 since sin2 u + cos2 u = 1 2 3 + cos2 u = 1 5 cos2 u = 1 − cos2 u = 9 25 16 25 cos u = ± 4 5 Since sin u > 0 and tan u > 0, u is in QI. Therefore cos u = 4 5 sin u = 3 5 csc u = 5 3 cos u = 4 5 sec u = 5 4 tan u = 3 4 cot u = 4 3 Simplify: sin x cos2 x − sin x sin x(cos2 x − 1) sin x(− sin2 x) − sin3 x Since sin2 x + cos2 x = 1 cos2 x − 1 = − sin2 x 2 c Marcia Drost, March 10, 2008 Simplify: sin θ cos θ + cos θ 1 + sin θ Verify the identity: cos θ sin θ + = csc θ 1 + cos θ sin θ (sin θ)(sin θ) + (cos θ)(1 + cos θ) = (1 + cos θ)(sin θ) sin2 θ + cos θ + cos2 θ = (1 + cos θ)(sin θ) sin2 θ + cos2 θ + cos θ = (1 + cos θ)(sin θ) 1 + cos θ = (1 + cos θ)(sin θ) 1 = sin θ csc θ = csc θ Factoring expressions: tan2 x − tan2 x · sin2 x tan2 x (1 − sin2 x) tan2 x (cos2 x) sin2 x · cos2 x 2 cos x sin2 x Factor: 4 tan2 θ + tan θ − 3 (4 tan θ − 3) (tan θ + 1) Factor: csc3 x − csc2 x − csc x + 1 csc2 x(csc x − 1) − 1(csc x − 1) (csc x − 1) (csc2 x − 1) (csc x − 1) (csc x − 1) (csc x + 1) c Marcia Drost, March 10, 2008 3 Simplify: sin t + cot t · cos t cos t · cos t sin t cos2 t sin t + sin t 2 sin t cos2 t + sin t sin t 2 sin t + cos2 t 1 = = csc t sin t sin t sin t + Rewrite: 1 so that it is not in fractional form. 1 + sin x 1 1 1 − sin x = · 1 + sin x 1 + sin x 1 − sin x 1 − sin x = 1 − sin2 x 1 − sin x = cos2 x 1 sin x = − cos2 x cos2 x sin x 1 1 − · = 2 cos x cos x cos x = sec2 x − tan x · sec x Rewrite: 1 1 − so that it is not in fractional form. 1 − sin x 1 + sin x 1 1 − , start with the 1 − sin x 1 + sin x RHS (right hand side) and simplify till it matches the LHS (left hand side). Note: To verify the identity 2 tan x · sec x = Verify: cos u = sec u + tan u 1 − sin u 4 c Marcia Drost, March 10, 2008 Use x = 2 tan θ, 0 < θ < √ √ π , to express 4 + x2 as a trigonometric function of θ. 2 p 4 + (2 tan θ)2 √ = 4 + 4 tan2 θ p = 4(1 + tan2 θ) p = 4 sec2 θ) 4 + x2 = = 2 sec θ Fundamental Trigonometric Identities Reciprocal Identities csc x = 1 sin x sec x = tan x = 1 cos x sin x cos x cot x = cot x = 1 tan x cos x sin x Pythagorean Identities sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x 1 + cot2 x = csc2 x Even-Odd Identities sin(−x) = − sin x cos(−x) = cos x tan(−x) = − tan x tan(90o − u) = cot u sec(90o − u) = csc u Cofunction Identities sin(90o − u) = cos u cos(90o − u) = sin u cot(90o − u) = tan u csc(90o − u) = sec u