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.
| a1 | a4 | ||
| a2 | a5 | ||
| a3 | a6 |
| b1 | |
| b2 |
| c1 | |
| c2 | |
| c3 |
| d1 | |
| d2 | |
| d3 |
| e1 | |
| e2 | |
| e3 |
* Note:
is 90° in radians.
If A is in degrees, use 90 instead of
For example:
This basically says that if two angles are supplementary (add to 180°) they have the same sine.
| f1 |
Or in degrees:
|
| g1 | |
| g2 | |
| g3 |
| h1 | |
| h2 | |
| h3 |
| j1 | |
| j2 | |
| j3 | |
| j4 | |
| j5 |
| k1 | |
| k2 | |
| k3 | |
| k4 |
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 |
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 |
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 |
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 |
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 |
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 |
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
mathopenref.com/trigidentities.html
Will link to the top of this page
mathopenref.com/trigidentities.html#c
Will link to this page with the page scrolled so that Section C is at the top
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 mathopenref.com/trigidentities.html#h2