Secant, Cosecant, and Cotangent Formulas
Learn the reciprocal trig functions: sec, csc, and cot.
Definitions, identities, and worked examples for each function.
The Formulas
csc(θ) = 1 / sin(θ) = hypotenuse / opposite
cot(θ) = 1 / tan(θ) = adjacent / opposite = cos(θ) / sin(θ)
The secant, cosecant, and cotangent are the reciprocals of cosine, sine, and tangent respectively. They appear frequently in calculus and advanced mathematics.
Variables
| Function | Reciprocal Of | Undefined When |
|---|---|---|
| sec(θ) | cos(θ) | cos(θ) = 0 (at 90°, 270°, etc.) |
| csc(θ) | sin(θ) | sin(θ) = 0 (at 0°, 180°, 360°, etc.) |
| cot(θ) | tan(θ) | sin(θ) = 0 (at 0°, 180°, 360°, etc.) |
Pythagorean Identities
1 + cot²(θ) = csc²(θ)
Example 1
Find sec(60°), csc(60°), and cot(60°).
cos(60°) = 0.5, sin(60°) = √3/2 ≈ 0.866, tan(60°) = √3 ≈ 1.732
sec(60°) = 1 / cos(60°) = 1 / 0.5 = 2
csc(60°) = 1 / sin(60°) = 1 / (√3/2) = 2/√3 = 2√3/3 ≈ 1.155
sec(60°) = 2, csc(60°) ≈ 1.155, cot(60°) = 1/√3 ≈ 0.577
Example 2
In a right triangle with opposite = 5 and hypotenuse = 13, find csc(θ) and cot(θ).
Adjacent = √(13² - 5²) = √(169 - 25) = √144 = 12
csc(θ) = hypotenuse / opposite = 13 / 5 = 2.6
cot(θ) = adjacent / opposite = 12 / 5 = 2.4
csc(θ) = 2.6, cot(θ) = 2.4
When to Use It
Use the reciprocal trig functions in these situations:
- Simplifying complex trigonometric expressions in calculus
- Solving trig equations that involve reciprocal functions
- Working with integrals and derivatives of trig functions
- Engineering formulas involving ratios of triangle sides