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Math 150 T17-Trig “Cheat Sheet” Page 1 Pythagorean Identities MATH 150 – TOPIC 17 TRIG “CHEAT SHEET” sin2 x + cos2 x = 1, 1 + tan2 x = sec2 x, 1 + cot2 x = csc2 x Definition of the Six Trigonometric Functions Right triangle definitions, where 0 < θ < π/2. Reduction Formulas sin(−x) = − sin x csc(−x) = − csc x sec(−x) = sec x Hypotenuse Opposite θ cos(−x) = cos x tan(−x) = − tan x cot(−x) = − cot x Adjacent sin θ = opposite hypotenuse csc θ = hypotenuse opposite cos θ = adjacent hypotenuse sec θ = hypotenuse adjacent cos(u ± v) = cos u cos v ∓ sin u sin v tan θ = opposite adjacent cot θ = adjacent opposite tan(u ± v) = Unit circle definitions, where θ is any angle. y 1 (x, y) • y 1= p 1 x • x2 θ 1 + y2 x y sin θ = = y 1 1 csc θ = y x cos θ = = 1 1 y tan θ = x 1 sec θ = x x cot θ = y Sum and Difference Formulas sin(u ± v) = sin u cos v ± cos u sin v tan u ± tan v 1 ∓ tan u tan v Cofunction Identities π sin − x = cos x 2 π csc − x = sec x 2 π − x = csc x sec 2 cos tan cot π 2 π 2 π 2 − x = sin x − x = cot x − x = tan x Double Angle Formulas Thus, x = cos θ, y = sin θ tan 2u = Reciprocal Identities 1 csc x 1 csc x = sin x sin x = 1 cos x 1 cos x = sec x sec x = 1 cot x 1 cot x = tan x tan x = Tangent and Cotangent Identities tan x = sin x cos x sin 2u = 2 sin u cos u cos 2u = cos2 u − sin2 u = 2 cos2 u − 1 = 1 − 2 sin2 u cot x = cos x sin x 2 tan u 1 − tan2 u Half Angle Formulas 1 − cos 2u 2 1 + cos 2u cos2 u = 2 1 − cos 2u tan2 u = 1 + cos 2u sin2 u = Continued on next page Math 150 T17-Trig “Cheat Sheet” Page 2 Product-to-Sum Formulas 1 sin u sin v = [cos(u − v) − cos(u + v)] 2 1 cos u cos v = [cos(u − v) + cos(u + v)] 2 1 sin u cos v = [sin(u + v) + sin(u − v)] 2 1 cos u sin v = [sin(u + v) − sin(u − v)] 2 Sum-to-Product Formulas u+v u−v sin u + sin v = 2 sin cos 2 2 u+v u−v sin u − sin v = 2 cos sin 2 2 u−v u+v cos cos u + cos v = 2 cos 2 2 u+v u−v cos u − cos v = −2 sin sin 2 2 y (− 12 , √ (− √ (− √ 2 2 , ) 2 2 • 3 1 , ) 2 2 • 5π 6 3π 4 √ (0, 1) 3 ) 2 • 2π 3 120 ◦ 60 135◦ 150◦ √ • 4 2 , − 22 ) 2 3 ) 2 π 4 ◦ 45◦ 30◦ • √ 2 2 , ) 2 2 ( • π 6 4π 3 • √ 1 3 (− 2 , − 2 ) 270◦ 3π 2 • 300 (0, −1) 5π 3 7π 4 • √ • x = cos θ y = sin θ √ 3 1 , ) 2 2 where θ is the indicated angle. 330◦ 11π 315◦ 6 • ◦ For any pair (x, y), √ ( 0◦ 0 ◦ 2π 360 210◦ 7π √ 225◦ • 6 3 1 (− 2 , − 2 ) 240◦ 5π (− 90◦ √ • π 3 180◦ (−1, 0) • π √ ( 12 , • π 2 ( ( • (1, 0) √ 3 , − 12 ) 2 √ 2 2 , − ) 2 2 √ ( 12 , − 23 ) x Previous Page Skills Assessment