Definitions of Tangent, Secant, and Cosecant

 

Let denote an angle.

Definition: The tangent of , denoted is

Definition: The secant of , denoted is

Definition: The cosecant of , denoted is

We summarize information about the domain and range of the six trig functions:

1. : Domain equals . Range equals [-1,1].
2. : Domain equals . Range equals [-1,1].
3. : Domain equals . Range equals .
4. : Domain equals . Range equals .
5. : Domain equals . Range equals .
6. : Domain equals . Range equals .

Using the alternate definition of sine and cosine as ratios of sides of right triangles, we can give alternate definitions of tangent, secant, cosecant, and cotangent.

1. .
2. .
3. .
4. .

Oscar Had A Heap Of Apples is a mnemonic to remember that , , and .

Starting with the known trigonometric identity we can derive two more identities by dividing both sides of this equation by and by .