Reciprocal Identities
Reference for secant, cosecant, and cotangent as reciprocals of cosine, sine, and tangent.
The Formulas
csc(θ) = 1 / sin(θ)
sec(θ) = 1 / cos(θ)
cot(θ) = 1 / tan(θ) = cos(θ) / sin(θ)
sec(θ) = 1 / cos(θ)
cot(θ) = 1 / tan(θ) = cos(θ) / sin(θ)
Each of the three main trig functions has a reciprocal counterpart. These reciprocal identities are used throughout calculus, physics, and engineering.
Variables
| Symbol | Meaning |
|---|---|
| csc(θ) | Cosecant — reciprocal of sine |
| sec(θ) | Secant — reciprocal of cosine |
| cot(θ) | Cotangent — reciprocal of tangent |
| θ | The angle (in degrees or radians) |
Example 1
Find csc(30°)
sin(30°) = 0.5
csc(30°) = 1 / sin(30°) = 1 / 0.5
= 2
Example 2
Find sec(60°)
cos(60°) = 0.5
sec(60°) = 1 / cos(60°) = 1 / 0.5
= 2
Example 3
Find cot(45°)
tan(45°) = 1
cot(45°) = 1 / tan(45°) = 1 / 1
= 1
When to Use Them
Use reciprocal identities when:
- Simplifying complex trig expressions
- Integrating or differentiating reciprocal trig functions
- Solving trig equations that involve sec, csc, or cot
- Converting between trig function forms