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Trigonometry (trig) identities

All these trig identities can be derived from first principles. But there are a lot of them and some are hard to remember. Print this page as a handy quick reference guide.

Recall that these identities work both ways. That is, if you have an expression that matches the left or right side of an identity, you can replace it with whatever is on the other side.

A. Reciprocal identities

a1 a4
a2 a5
a3 a6

B. Ratio identities

b1
b2

C. Opposite Angle identities

c1
c2 (Caution - not like c1, c3)
c3

D. Pythagorean identities

d1
d2
d3

E. Complementary angle identities

e1
e2
e3

* Note: is 90° in radians.
If A is in degrees, use 90 instead of
For example: 

F. Supplementary angle identities

This basically says that if two angles are supplementary (add to 180°) they have the same sine.

f1 where
Or in degrees:
where

G. The sum identities

g1
g2
g3

H. The difference identities

h1
h2
h3

J. The double angle identities

j1
j2
j3
j4
j5

K. The half angle identities

k1
k2
k3
k4

M. The sine identities

These show how to represent the sine function in terms of the other five functions. Some of these identities may also appear under other headings.

m1
m2
m3
m4
m5

N. The cosine identities

These show how to represent the cosine function in terms of the other five functions. Some of these identities may also appear under other headings.

n1
n2
n3
n4
n5

P. The tangent identities

These show how to represent the tangent function in terms of the other five functions. Some of these trig identities may also appear under other headings.

p1
p2
p3
p4
p5

Q. The cosecant identities

These show how to represent the cosecant function in terms of the other five functions. Some of these trig identities may also appear under other headings.

q1
q2
q3
q4
q5

R. The secant identities

These show how to represent the secant function in terms of the other five functions. Some of these trig identities may also appear under other headings.

r1
r2
r3
r4
r5

S. The cotangent identities

These show how to represent the cotangent function in terms of the other five functions. Some of these identities may also appear under other headings.

s1
s2
s3
s4
s5

Linking to this page

If you would like to make links to this page, you can customize the link so it points to a particular identity or group of identities. The link should be the URL of the page, followed by a #, follwed by the section or identity you want. For example

http://www.mathopenref.com/trigidentities.html Will link to the top of this page

http://www.mathopenref.com/trigidentities.html#c Will link to this page with the page scrolled so that Section C is at the top

http://www.mathopenref.com/trigidentities.html#h2 Will link to this page with the page scrolled so that the identity h2 is at the top and outlined.

For example http://www.mathopenref.com/trigidentities.html#h2