Trig Identities

AP PreCalculus BC

pythagorean identities

sin²(theta)+cos²(theta)=1

tan²(theta)+1=sec²(theta)

1+cot²(theta)=csc²(theta)

cos(a+b)

cos(a)cos(b)-sin(a)sin(b)

sin(a-b)

sin(a)cos(a)-cos(a)sin(b)

tan(A+B)

(tanA+tanB)/(1-tanAtanB)

sin(2theta)

2sin(theta)cos(theta)

cos(2theta)

cos²(theta)-sin²(theta)

1-2sin²(theta)

2cos²(theta)-1

tan(2theta)

2tan(theta)/(1-tan²(theta))

sin(theta/2)

plus or minus √((1-cos(theta))/2)

cos(theta/2)

plus or minus √((1+cos(theta))/2)

tan(theta/2)

plus or minus √(1-cos(theta))/(1+cos(theta))

double angle identities

sin(2theta)=2sin(theta)cos(theta)

cos(2theta)=cos²(theta)-sin²(theta)

cos(2theta)=1-2sin²(theta)

cos(2theta)=2cos²(theta)-1

tan(2theta)=2tan(theta)/(1-tan²(theta))

half-angle identities

sin(theta/2)=plus or minus √((1-cos(theta))/2)

cos(theta/2)=plus or minus √((1+cos(theta))/2)

tan(theta/2)=plus or minus √(1-cos(theta))/(1+cos(theta))

inverse trig regions

sin^-1x, -pi/2 ≤ x ≤ pi/2

cos^-1x, 0 ≤ x ≤ pi

tan^-1x, -pi/2 < x < pi/2