Definitions of Trig Functions of Any Angle TRIGONOMETRY Let be any angle in the standard position Right Triangle Definition of Trig Functions and let P(,) x y be a point on the terminal side of . If r is the distance from 0,0 toxy, and r x22 y 0, x 0, y 0 y x y sin , cos , tan r r x opposite a hypotenuse c r r x sin csc csc , sec , cot hypotenuse c opposite a y x y adjacent b hypotenuse c cos sec Reference Angles hypotenuse c adjacent b Let be an angle in standard position. Its reference opposite a adjacent b angle is the acute angle formed by the terminal tan cot adjacent b opposite a side of and the horizontal axis. Degrees to Radians Formulas Example: If x is an angle in degrees and t is an angle in 315 radians then: 360 315 45 t x180 t tx and 180x 180 Sines, Cosines, and Tangents of Special Angles The Signs of Trig Functions Q I Q II Q III Q IV sin + + cos + + tan + + Formulas and Identities sin cos Ratio: tan cot cos sin Reciprocal Identities Even/Odd 1 cos( ) cos Even 0 csc 30 45 60 90 sin sin( ) sin 6 4 3 2 1 Odd sin 0 1 1 tan( ) tan 2 3 sec 2 2 2 cos 1 cos 1 0 cot tan 0 dne tan 3 1 3 3 Pythagorean Identities sin2 cos 2 1 tan 2 1 sec 2 1 cot22 csc TRIGONOMETRY Formulas and Identities Double Angle Formulas Half-Angle Formulas sin(2 ) 2sin cos 1 cos Graphs of Trig Functions sin cos(2 ) cos22 sin 22 yx sin 2 1 cos 2cos 1 cos Domain: x, 22 1 2sin2 Range: y [ 1,1] 1 cos sin 2 tan tan Period: 2 tan(2 ) 2 sin 1 cos 1 tan2 Amplitude: 1 yx cos Cofunction Formulas Domain: sin(90 ) cos cos(90 ) sin Range: Period: sin cos cos sin 22 Amplitude: 1 Sum and Difference Formulas yx tan sin( ) sin cos cos sin Domain: cos( ) cos cos sin sin x and xn tan tan 2 tan( ) 1 tan tan Range: y, Period: Sum to Product Formulas Amplitude: None sin sin 2sin cos 22 yx cot Domain: x and xn sin sin 2cos sin 22 Range: Period: cos cos 2cos cos 22 Amplitude: None cos cos 2sin sin 22 yx sec Domain: Product to Sum Formulas 1 sin sin cos( ) cos( ) 2 Range: 1 cos cos cos( ) cos( ) y ( , 1] [1, ) 2 Period: 2 1 sin cos sin( ) sin( ) Amplitude: None 2 1 cos sin sin( ) sin( ) yx csc 2 Domain: Inverse Trig Functions Range: ysin1 x is equivalent to x sin y , y 22 Period: ycos1 x is equivalent to x cos y , 0 y Amplitude: None ytan1 x is equivalent to x tan y , y 22 .