Reciprocal Identities
Reference for the six reciprocal trig identities: sec, csc, and cot as reciprocals of cos, sin, and tan.
Includes definitions and domain restrictions.
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
Domain Restrictions
- csc(θ) = 1/sin(θ) is undefined where sin(θ) = 0 — at 0°, 180°, 360°, and all multiples of 180°
- sec(θ) = 1/cos(θ) is undefined where cos(θ) = 0 — at 90°, 270°, and all odd multiples of 90°
- cot(θ) = 1/tan(θ) is undefined where tan(θ) = 0 — at 0°, 180°, and all multiples of 180°
- Always check domain restrictions when solving equations involving sec, csc, or cot — extraneous solutions can appear at excluded angles
Key Notes
- Definitions: csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ = cos θ/sin θ. These three functions are simply the multiplicative inverses of the three primary trig functions — not the same as the inverse trig functions (arcsin, arccos, arctan).
- Domain restrictions: csc θ and cot θ are undefined where sin θ = 0 (at θ = 0°, 180°, 360°, …). sec θ and tan θ are undefined where cos θ = 0 (at θ = 90°, 270°, …). Vertical asymptotes appear at these points in the graphs.
- Extended Pythagorean identities: 1 + cot²θ = csc²θ; and tan²θ + 1 = sec²θ. Both are derived by dividing sin²θ + cos²θ = 1 by sin²θ or cos²θ respectively. Useful for simplifying expressions that involve sec or csc.
- Non-obvious integrals: ∫sec θ dθ = ln|sec θ + tan θ| + C; ∫csc θ dθ = −ln|csc θ + cot θ| + C. These results are frequently needed in calculus and are not derivable by simple inspection — they require a clever substitution.
- Applications: Reciprocal identities appear in electrical engineering (phasor impedance calculations using sec and csc), optics (the critical angle formula involves sec), Fourier analysis, and advanced integration problems in physics and engineering.