Trig Cheat Sheet - Lamar University

1 trig Cheat Sheet Definition of the trig Functions Right triangle definition For this definition we assume that Unit circle definition p For this definition q is any angle. 0 < q < or 0 < q < 90 . 2 y ( x, y ). hypotenuse 1. y q opposite x x q adjacent opposite hypotenuse y 1. sin q = csc q = sin q = =y csc q =. hypotenuse opposite 1 y adjacent hypotenuse x 1. cos q = sec q = cos q = = x sec q =. hypotenuse adjacent 1 x opposite adjacent y x tan q = cot q = tan q = cot q =. adjacent opposite x y Facts and Properties Domain The domain is all the values of q that Period can be plugged into the function. The period of a function is the number, T, such that f (q + T ) = f (q ) . So, if w sin q , q can be any angle is a fixed number and q is any angle we cos q , q can be any angle have the following periods.

2 1 . tan q , q n + p , n = 0, 1, 2,K. 2 2p sin ( wq ) T=. csc q , q n p , n = 0, 1, 2,K w 1 2p sec q , q n + p , n = 0, 1, 2,K cos (wq ) T =. 2 w cot q , q n p , n = 0, 1, 2,K p tan (wq ) T =. w Range 2p csc (wq ) T =. The range is all possible values to get w out of the function. 2p -1 sin q 1 csc q 1 and csc q -1 sec (wq ) T =. w -1 cos q 1 sec q 1 and sec q -1 p - < tan q < - < cot q < cot (wq ) T =. w 2005 Paul Dawkins Formulas and Identities Tangent and Cotangent Identities Half Angle Formulas (alternate form). sin q cos q q 1 - cos q 1. tan q = cot q = sin = sin 2 q = (1 - cos ( 2q ) ). cos q sin q 2 2 2. Reciprocal Identities q 1 + cos q 1. csc q =. 1. sin q =. 1 cos 2. = . 2. cos 2 q =. 2. (1 + cos ( 2q ) ). sin q csc q 1 1 q 1 - cos q 1 - cos ( 2q ). sec q = cos q = tan = tan 2 q =.

3 Cos q sec q 2 1 + cos q 1 + cos ( 2q ). 1 1 Sum and Difference Formulas cot q = tan q =. tan q cot q sin (a b ) = sin a cos b cos a sin b Pythagorean Identities cos (a b ) = cos a cos b m sin a sin b sin 2 q + cos 2 q = 1. tan a tan b tan 2 q + 1 = sec 2 q tan (a b ) =. 1 m tan a tan b 1 + cot 2 q = csc 2 q Product to Sum Formulas 1. Even/Odd Formulas sin a sin b = cos (a - b ) - cos (a + b ) . sin ( -q ) = - sin q csc ( -q ) = - csc q 2. 1. cos ( -q ) = cos q sec ( -q ) = sec q cos a cos b = cos (a - b ) + cos (a + b ) . 2. tan ( -q ) = - tan q cot ( -q ) = - cot q 1. sin a cos b = sin (a + b ) + sin (a - b ) . Periodic Formulas 2. If n is an integer. 1. cos a sin b = sin (a + b ) - sin (a - b ) . sin (q + 2p n ) = sin q csc (q + 2p n ) = csc q 2. Sum to Product Formulas cos (q + 2p n ) = cos q sec (q + 2p n ) = sec q a + b a - b.

4 Tan (q + p n ) = tan q cot (q + p n ) = cot q sin a + sin b = 2sin cos . 2 2 . Double Angle Formulas a + b a - b . sin a - sin b = 2 cos sin . sin ( 2q ) = 2sin q cos q 2 2 . cos ( 2q ) = cos 2 q - sin 2 q cos a + cos b = 2 cos . a + b a - b . cos . = 2 cos 2 q - 1 2 2 . a + b a - b . = 1 - 2sin 2 q cos a - cos b = -2sin sin . 2 2 . 2 tan q tan ( 2q ) = Cofunction Formulas 1 - tan 2 q p p . Degrees to Radians Formulas sin - q = cos q cos - q = sin q 2 2 . If x is an angle in degrees and t is an angle in radians then p p . csc - q = sec q sec - q = csc q p t px 180t 2 2 . = t= and x = p p . 180 x 180 p tan - q = cot q cot - q = tan q 2 2 . 2005 Paul Dawkins Unit Circle y ( 0,1). p 1 3 . 1 3 2 , 2 . - , 2 . 2 2 p 2 2 . 2p 90 , . 2 2 3 2 2 . - , 3. 2 2 120 p 3p 60 3 1 . 4 2 , 2 . 3 1 4 45 p.

5 - , 135 . 2 2 5p 6. 6 30 . 150 . ( -1,0 ) p 180 0 0 (1,0 ). 360 2p x 210 . 7p 330 . 11p 6 225 . 3 1 6 3 1 . - ,- 5p 315 ,- . 2 2 2 2 . 4 240 300 7p 2 4p 270 . - 2. ,- 5p 4 2 2 . 2 2 3 3p . 2. ,- 2.. 3 . 1 3 2 . - ,- 1 3 . 2 2 ,- . 2 2 . ( 0,-1). For any ordered pair on the unit circle ( x, y ) : cos q = x and sin q = y Example 5p 1 5p 3. cos = sin =- 3 2 3 2. 2005 Paul Dawkins Inverse trig Functions Definition Inverse Properties y = sin -1 x is equivalent to x = sin y cos ( cos -1 ( x ) ) = x cos -1 ( cos (q ) ) = q y = cos -1 x is equivalent to x = cos y sin ( sin -1 ( x ) ) = x sin -1 ( sin (q ) ) = q y = tan -1 x is equivalent to x = tan y tan ( tan -1 ( x ) ) = x tan -1 ( tan (q ) ) = q Domain and Range Function Domain Range Alternate Notation p p sin -1 x = arcsin x y = sin -1 x -1 x 1 - y.

6 2 2 cos -1 x = arccos x y = cos -1 x -1 x 1 0 y p tan -1 x = arctan x p p y = tan -1 x - < x < - < y<. 2 2. Law of Sines, Cosines and Tangents c b a a g b Law of Sines Law of Tangents sin a sin b sin g = = a - b tan 12 (a - b ). =. a b c a + b tan 12 (a + b ). Law of Cosines b - c tan 12 ( b - g ). =. a 2 = b2 + c 2 - 2bc cos a b + c tan 12 ( b + g ). b 2 = a 2 + c 2 - 2ac cos b a - c tan 12 (a - g ). =. c = a + b - 2ab cos g a + c tan 12 (a + g ). 2 2 2. Mollweide's Formula a + b cos 12 (a - b ). =. c sin 12 g 2005 Paul Dawkins