- 1 Basic Definitions
- 2 Even-Odd Identities
- 3 Reciprocal Relations
- 4 Pythagorean Identities
- 5 Angle Addition/Subtraction Identities
- 6 Double Angle Identities
- 7 Further Conclusions
The six basic trigonometric functions can be defined using a right triangle:
The six trig functions are sine, cosine, tangent, cosecant, secant, and cotangent. They are abbreviated by using the first three letters of their name (except for cosecant which uses ). They are defined as follows:
Based on the above identities, we can also claim that
This is only true when is in the domain of .
From the first section, it is easy to see that the following hold:
Another useful identity that isn't a reciprocal relation is that .
Note that ; the former refers to the inverse trigonometric functions.
Using the Pythagorean Theorem on our triangle above, we know that . If we divide by we get , which is just . Dividing by or instead produces two other similar identities. The Pythagorean Identities are listed below:
(Note that the last two are easily derived by dividing the first by and , respectively.)
Angle Addition/Subtraction Identities
Once we have formulas for angle addition, angle subtraction is rather easy to derive. For example, we just look at and we can derive the sine angle subtraction formula using the sine angle addition formula.
We can prove easily by using and .
Double Angle Identities
Double angle identities are easily derived from the angle addition formulas by just letting . Doing so yields:
We can see from the above that
Half Angle Identities
Using the double angle identities, we can derive half angle identities. The double angle formula for cosine tells us . Solving for we get where we look at the quadrant of to decide if it's positive or negative. Likewise, we can use the fact that to find a half angle identity for sine. Then, to find a half angle identity for tangent, we just use the fact that and plug in the half angle identities for sine and cosine.
(Otherwise known as sum-to-product identities)
Law of Sines
- Main article: Law of Sines
The extended Law of Sines states
Law of Cosines
- Main article: Law of Cosines
The Law of Cosines states
Law of Tangents
- Main article: Law of Tangents
The Law of Tangents states that if and are angles in a triangle opposite sides and respectively, then
A further extension of the Law of Tangents states that if , , and are angles in a triangle, then
- (This is also written as )
The two identities right above here were based on identities others posted on this site with a substitution.