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Definitions of Trig Functions of Any Angle TRIGONOMETRY Let be any angle in the standard position and let P ( x, y ) be a point on the terminal side of . If r is the distance from 0, 0 Right Triangle Definition of Trig Functions to x, y and r x 2 y 2 0, x 0, y 0 y , r r csc , y x cos , r r sec , x sin opposite a hypotenuse c adjacent b cos hypotenuse c opposite a tan adjacent b sin hypotenuse c opposite a hypotenuse c sec adjacent b adjacent b cot opposite a csc y x x cot y tan Reference Angles Let be an angle in standard position. Its reference angle is the acute angle formed by the terminal side of and the horizontal axis. Degrees to Radians Formulas If x is an angle in degrees and t is an angle in Example: 315 radians then: 360 315 45 t x 180t t and x 180 x 180 Sines, Cosines, and Tangents of Special Angles The Signs of Trig Functions sin cos tan QI + + + Q II + Q III + Q IV + Formulas and Identities sin cos cot Ratio: tan cos sin Reciprocal Identities Even/Odd 0 30 sin 0 cos 1 tan 3 2 0 3 3 1 2 6 45 2 2 2 2 1 4 60 3 2 1 2 3 90 3 1 0 2 1 sin 1 sec cos 1 cot tan csc cos( ) cos sin( ) sin Odd tan( ) tan dne Pythagorean Identities sin 2 cos 2 1 tan 2 1 sec 2 1 cot 2 csc 2 Even TRIGONOMETRY Formulas and Identities Double Angle Formulas sin(2 ) 2sin cos Graphs of Trig Functions cos(2 ) cos 2 sin 2 y sin x Domain: x , Range: y [ 1,1] Period: 2 Amplitude: 1 y cos x Domain: x , Range: y [ 1,1] Period: 2 Amplitude: 1 y tan x Domain: x and x n 2 Range: y , Period: Amplitude: None y cot x Domain: x and x n Range: y , Period: Amplitude: None y sec x Domain: x and x 1 2sin 2 tan(2 ) 2 tan 1 tan 2 Cofunction Formulas sin(90 ) cos sin 2 2 1 cos 2 2 1 cos sin tan 2 sin 1 cos cos cos(90 ) sin sin cos cos sin 2 2 Sum and Difference Formulas sin( ) sin cos cos sin cos( ) cos cos sin sin tan tan tan( ) 1 tan tan Sum to Product Formulas sin sin 2sin cos 2 2 sin sin 2 cos sin 2 2 cos cos 2 cos cos 2 2 cos cos 2sin sin 2 2 Product to Sum Formulas 2 n Range: y (, 1] [1, ) Period: 2 Amplitude: None y csc x Domain: x Range: 2 cos 2 1 Half-Angle Formulas 1 cos and x n y (, 1] [1, ) Period: 2 Amplitude: None 1 cos( ) cos( ) 2 1 cos cos cos( ) cos( ) 2 1 sin cos sin( ) sin( ) 2 1 cos sin sin( ) sin( ) 2 sin sin Inverse Trig Functions y sin 1 x is equivalent to x sin y, y y cos x is equivalent to x cos y, 2 2 0 y y tan 1 x is equivalent to x tan y, 1 2 y 2