Table of trig identities

Negative Angle Identities
sin(- ! ) = - sinθ
cos(- ! ) = cosθ
tan(- ! ) = - tanθ
csc(- ! ) = - cscθ
sec(- ! ) = secθ
cot(- ! ) = - cotθ
Addition and Subtraction Identities
Co-function Identities If A + B = π2
sin A = cos B
sec A = csc B
tan A = cot B
Quotient Identities
1
sin !
1
secθ =
cos !
sin (A + B) =
sin A cos B + cos 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
Pythagorean Identities
sin (A – B) =
sin A cos B – cos A sin B
sin θ + cos θ = 1
cos (A – B) =
cos A cos B + sin A sin B
1 + tan θ = sec θ
tan A – tan B
1 + tan A tan B
tan (A – B) =
Double Angle Identities
cscθ =
2
tanθ
=
cotθ =
2
2
1 + cot θ = csc2θ
Half-Angle Identities
=
±
1 – cos !
2
2
cos
!
2
=
±
1 + cos !
2
= 2cos θ -- 1
tan
!
2
=
±
1 – cos !
1 + cos !
cos 2θ
= cos θ -- sin !
2
2
cos !
sin !
2
!
2
= 2sinθ cosθ
=
2
sin
sin 2 θ
1
tan !
= 1 – 2sin2θ
tan 2 !
=
2tan !
2
1 – tan !
Product Identities
sinAcosB
=
cosAsinB
=
cosAcosB
=
sinAsinB
=
( sin (A + B) + sin (A – B) )
1
( sin (A + B) – sin (A – B) )
2
1
( cos (A + B) + cos (A – B) )
2
1
( cos (A – B) – cos (A + B) )
2
1
2
Sum Identities
sinA + sinB
sinA – sinB
cosA + cosB
cosA – cosB
( A +2 B
A+B
= 2cos(
2
A+B
= 2cos(
2
A+B
= - 2sin(
2
=
2sin
)cos( A 2– B
)sin( A 2– B
)cos( A 2– B
)sin( A 2– B
)
)
)
)
THE UNIT CIRCLE
(0,1)
!
( ,
1
2
( ,
2
2
( , )
1
2
3
2
(- 1,0)
( ,
3
2
2
2
)
3
2
3!
4
2
)
!
2!
3
( ,
1
2
3
!
3
2
)
( , )
2
2
4
5!
6
( , )
!
3
2
6
1
2
0 (1,0)
!
1
2
2
2
)
7!
6
( , )
2
2
2
2
11!
6
5!
4
( ,
1
2
7!
4
4!
3
3
2
)
5!
3
( ,
1
2
3!
2
3
2
( , )
3
2
1
2
( , )
2
2
2
2
)
(0,- 1)
sin ! =
TRIGONOMETRIC DEFINITIONS
SPECIAL ANGLE TRIANGLES
opposite
y
=
r hypotenuse
2
adjacent
x
cos ! = =
r hypotenuse
tan ! =
y opposite
=
x adjacent
θ
x
1
30°
1
3
or if it is the unit circle,
r=1 so
45°
60°
1
2
2
1
2
30°
45°
y
60°
2
1
45°
1
r
45°
2
2
3
2