Trigonometry

Trigonometric Functions

Sine

sinθ =

Opposite Hypotenuse

Cosine

cosθ =

Adjacent Hypotenuse

Tangent

tanθ =

Opposite Adjacent

Cotangent

cotθ =

Adjacent Opposite

Secant

secθ =

Hypotenuse Adjacent

Cosecant

cosecθ =

Hypotenuse Opposite

Trigonometric Identities

sinθ =

1 cosecθ

cosθ =

1 secθ

tanθ =

1 cotθ

=

sinθ cosθ

Pythagorean Identities

sin²θ + cos²θ = 1

tan²θ + 1 = sec²θ

cot²θ + 1 = cosec²θ

Opposite Angle Identities

sin(− θ) = − sinθ

cos(− θ) = cosθ

tan(− θ) = − tanθ

Double Angle Identities

sin(2θ)	=	2 sinθ cosθ
	=	2 tanθ 1 + tan²θ

cos(2θ)	=	cos²θ − sin²θ
	=	2 cos²θ − 1
	=	1 − 2 sin²θ
	=	1 − tan²θ 1 + tan²θ

tan(2θ)

=

2 tanθ 1 − tan²θ

Half Angle Identities

sin (

θ 2

)

=

±

1 − cos θ 2

cos (

θ 2

)

=

±

1 + cos θ 2

tan ( θ 2 )	=	± 1 − cos θ 1 + cos θ
	=	sin θ 1 + cos θ
	=	1 − cos θ sin θ
	=	cosec θ − cot θ

cot ( θ 2 )	=	± 1 + cos θ 1 − cos θ
	=	sin θ 1 − cos θ
	=	1 + cos θ sin θ
	=	cosec θ + cot θ

Angle Sum and Difference Identities

sin (A + B)	=	sin(A) cos(B) + cos(A) sin(B)
sin (A − B)	=	sin(A) cos(B) − cos(A) sin(B)

cos (A + B)	=	cos(A) cos(B) − sin(A) sin(B)
cos (A − B)	=	cos(A) cos(B) + sin(A) sin(B)

tan (A + B)	=	tan(A) + tan(B) 1 − tan(A) tan(B)
tan (A − B)	=	tan(A) − tan(B) 1 + tan(A) tan(B)