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Definitions of Trig Functions of Any

TRIGONOMETRY Let  be any angle in the standard position Right 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   toxy,  and r x22  y 0, x  0, y  0

y x y sin , cos   , tan   r r x

opposite a c r r x sin csc  csc , sec   , cot   hypotenuse c opposite a y x y adjacent b hypotenuse c cos sec  Reference 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 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 1 tan( )   tan sin 0 2 3 1 sec   2 2 2 cos 1 cos 1 0 cot  tan tan 0 3 1 3 dne 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:  ysin1 x is equivalent to x  sin y ,   y  22 Period: ycos1 x is equivalent to x  cos y , 0  y   Amplitude: None  ytan1 x is equivalent to x  tan y ,   y  22