Trigonometric Identities

Trigonometric Identitiessin2 x + cos2 x = 12 21+ tan x = sec x2 21+ cot x = csc xsin x = cos( 90− x) = sin( 180−x)cos x = sin( 90− x) = − cos( 180−x)tan x = cot( 90− x) = − tan( 180−x)Angle-sum and angle-difference formulassin( a ± b) = sin a cosb ± cosa sinbcos( a ± b) = cosa cosb m sin a sinbtan a ± tanbtan( a ± b)=1m tan a tanbcot a cotbm1cot( a ± b)=cotb± cot asin( a + b)sin( a − b) = sin 2 a − sin 2 b = cos 2 b − cos2 acos( a + b) cos( a − b) = cos 2 a − sin 2 b = cos 2 b − sin2 aDouble-angle relations2 tan asin 2a = 2sin a cosa=21 + tan a2 2 2 2 1cos2a = cos a − sin a = 2cos a − 1= 1− 2 sin a = − 1+22 tan acot a −1tan 2a= cot 2a=21 − tan a2 cot atantan22aaMultiple-angle relations3sin 3a = 3sin a − 4sina33tana − tan acos3a = 4cos 3 a − 3cosatan 3a=21−3tana3sin 4a = 4sin a cosa − 8sin a cosa34 24 tan a − 4 tan acos4a = 8cos a − 8cosa + 1tan 4a=2 41− 6tana + tan asin5a = 5sin a − 20sin 3 a + 16sin5 acos5a = 16cos 5 a − 20cos 3 a + 5cosasin 6a = 32 cos 5 a sin a − 32 cos 3 sin a + 6cosa sin acos6a = 32cos 6 a − 48cos 4 a + 18cos2 a −1

tan( n − 1) a + tan asin na = 2 sin( n −1) a cosa − sin( n − 2)a tan na =1−tan( n −1) a tan acosna = 2 cos( n −1)cosa − cos( n − 2)aFunction-product relations1sin a sinb = cos( a − b) − cos( a + b)b 2 g12 g12 g12 gcosa cosb = cos( a − b) + cos( a + b)sin a cosb = sin( a + b) + sin( a − b)cosa sinb = sin( a + b) − sin( a − b)Function-sum and function-difference relationsFa + bI a − bsin a + sinb= sin cosHG K J F I2HG K J2 2Fa + bI a − bsin a − sinb= cos sinHG K J F I2HG K J2 2Fa + bI a − bcosa+ cosb= cos cosHG K J F I2HG K J2 2F a + bI a − bcosa− cosb= − sin sinHG K J F I2HG K J2 2sin( a + b)tan a + tanb=cosacosbsin( a − b)tan a − tanb=cosacosbHalf-angle relationsa 1 cosasin = ± − a 1cos = ± +2 22a 1 cosatan = ± − 2 1+cosa1 cosasin a= − =sin a 1 + cosaa 1 cosa1 cosasin acot = ± + = + =2 1−cosasin a 1 − cosacosa2

Trig functions of special anglesAngle sin cos tan0 0 1 0151830364554607224122d 3 −1i 4d 3 + 1i 2 − 35 −15+ 52 5 −142 22 5+5325− 55 + 15 −1 5−52 242 222225 + 15− 55 + 1 242 22 5−532125+ 55 −15 + 1 5−52 242 22753 + 13 −1+4490 1 0 …d1dd332d i d i 2 33diiii