Secant, Cosecant, and Cotangent Formulas
Reciprocal trig functions: sec = 1/cos, csc = 1/sin, cot = 1/tan.
Covers definitions, Pythagorean identities sec²−tan²=1 and csc²−cot²=1, with worked examples.
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
Key Notes
- Definitions: sec θ = 1/cosθ; csc θ = 1/sinθ; cot θ = cosθ/sinθ = 1/tanθ. These are the reciprocal trig functions — not the same as inverse trig functions (arcsin, arccos, arctan). "sec" is the reciprocal of cosine, NOT the inverse.
- Graph behavior: sec θ and csc θ have period 2π, range (−∞, −1] ∪ [1, ∞) — they are never between −1 and 1. cot θ has period π (same as tan), range all real numbers. All three have vertical asymptotes where their respective denominators equal zero.
- Pythagorean identities from reciprocals: Dividing sin²θ + cos²θ = 1 by cos²θ gives tan²θ + 1 = sec²θ. Dividing by sin²θ gives 1 + cot²θ = csc²θ. These allow substitution between sec, tan, csc, and cot in integration and simplification.
- Key integrals: ∫sec θ dθ = ln|sec θ + tan θ| + C; ∫csc θ dθ = −ln|csc θ + cot θ| + C; ∫sec²θ dθ = tan θ + C; ∫csc²θ dθ = −cot θ + C. The first two are non-obvious and frequently appear in calculus courses and physics problems.
- Applications: These functions appear in beam deflection equations (sec and csc in Euler column buckling), electrical engineering (phasor impedance), calculus integrals involving trigonometric substitution, and navigation (the Mercator projection uses a secant function for latitude scaling).