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1 Right Triangle: sin θ opp hyp Right Triangle: cos θ adj hyp Right Triangle: tan θ Circular function, radius r sin θ Circular function, radius r cos θ opp adj y r x r 2 Circular function, radius r tan θ Reciprocal Identity: csc θ Reciprocal Identity: sec θ Reciprocal Identity: cot θ Reciprocal Identity: 1 y x 1 sin θ 1 cos θ 1 tan θ sin x csc x or cos x sec x or tan x cot x 3 Pythagorean Identities: sin 2 x + cos 2 x = 1 Pythagorean Identities: 2 1 + tan x = sec 2 x Pythagorean Identities: 2 1 + cot x = csc 2 x Cofunction Identities sin (π/2 - x) cos x Cofunction Identities cos (π/2 - x) sin x 4 Cofunction Identities tan (π/2 - x) Cofunctions of cot x are equal. Complementary Angles Sum/Difference Formulas: sin (u + v) = sinu cosv ± sinv cosu Sum/Difference Formulas: cos (u + v) = Sum/Difference Formulas: tan (u + v) = cosu cosv sinu sin v tan u ± tan v 1 tan u tan v 5 Double angle Formula sin 2u 2 sin u cos u Double angle Formula cos2u - sin2u 2 cos2u - 1 1 - 2 sin2u cos 2u Double angle Formula tan 2u 2 tan u 1 − tan2 u Power Reducing Formulas 2 sin u 1 − cos2u 2 Power Reducing Formulas 2 cos u 1 + cos2u 2 6 Graph of y = sin x Graph of y = cos x Domain (-∞,∞) Range [-1,1] Period 2π Domain (-∞,∞) Range [-1,1] Period 2π Domain Graph of y = tan x x≠ nπ , n∍I 2 Range (-∞,∞) Period π Domain Graph of y = cot x x≠ nπ , n∍I 2 Range (-∞,∞) Period π Domain Graph of y = sec x x≠ nπ , n∍I 2 Range |y| > 1 Period 2π 7 Power Reducing Formulas 2 tan u 1 − cos2u 1 + cos2u Product-to-Sum: sin u sin v 1 [cos(u − v) − cos(u + v)] 2 Product-to-Sum: cos u cos v Product-to-Sum: sin u cos v Product-to-Sum: cos u sin v 1 [cos(u − v) + cos(u + v)] 2 1 [sin(u + v) + sin(u − v)] 2 1 [sin(u + v) − sin(u − v)] 2 8 Domain Graph of y = csc x Graph of y = sin-1 x Graph of y = cos-1 x Graph of y = tan-1 x x ≠ nπ , n ∍ I Range |y| > 1 Period 2π Domain [-1,1] Range ⎛ π π⎞ ⎜− ' ⎟ ⎝ 2 2⎠ Domain [-1,1] Range [0,π] Domain (-∞,∞) Range ⎛ π π⎞ ⎜− , ⎟ ⎝ 2 2⎠ Graph of y = sec-1 x Domain |x| >1 Range [0,π], y ≠ π 2 9 Graph of y = csc-1 x Domain |x| > 1 Range ⎛ π π⎞ ⎜− , ⎟, y ≠ 0 ⎝ 2 2⎠ Graph of y = cot-1 x Domain (-∞,∞) Range (0,π) Graph of y = ex Domain (-∞,∞) Range (0,∞) Graph of y = ln x Domain (0,∞) Range (-∞,∞) Quadratic Formula −b ± b 2 − 4ac x= 2a 10 Law of Sines Sin A Sin B SinC = = a b c Law of Cosines c 2 = a 2 + b 2 − 2abcosC Pythagorean Theorem sec −1 x −1 csc x c2 = a 2 + b 2 1 cos x −1 1 sin x −1 11 a ⋅a x y x a y a (a ) x y a −x log a x + log a y a a x+y x−y a xy 1 x a log a xy 12 sin −1 x + cos−1 x π 2 tan −1 x + cot −1 x π 2 csc −1 x + sec −1 x π 2 log a x − log a y ⎛ x⎞ log a ⎜⎜ ⎟⎟ ⎝ y⎠ log a x b b log a x 13 Change of Base Formula log a x Transformation of Trigonometric Graphs ln x ln a Vertical stretch or shrink: reflection about x-axis Vertical Shift y = af (b(x + c)) + d y = af (b(x + c)) + d Horizontal stretch or shrink: Horizontal Shift reflection about y-axis Point-Slope Equation y = m( x − x1 ) + y1 Slope General Linear Equation rise Δy y2 − y1 m= = = run Δx x2 − x1 Ax + By = C 14 2 30°-60°-90° Right Triangle 60° 30° 3 45°-45°-90° Right Triangle 45° 2 1 45° 1 1