How to Use Secant Calculator
The Secant Calculator computes the secant of any angle with a simple two-step workflow. Select the angle unit and enter the angle value.
- Select the unit — Choose Degrees or Radians depending on how your angle is expressed.
- Enter the angle θ — Type any real number. For degrees, common values include 0, 30, 45, 60, 120; for radians, values like π/3 ≈ 1.047.
- Review the result — The Secant Calculator shows sec(θ), the intermediate cos(θ) value, the substitution formula, and the angle in both units.
If cos(θ) = 0 at the entered angle, the calculator displays a clear undefined message instead of a numeric result.
Formula & Theory - Secant Calculator
The Secant Calculator uses this core formula or rule based on the reciprocal identity:
sec(θ) = 1 / cos(θ)
Defined for: cos(θ) ≠ 0
Undefined at: θ = 90° + n·180° (n = 0, ±1, ±2, ...)
θ = π/2 + n·π (in radians)| Symbol | Meaning |
|---|---|
| θ | Input angle |
| cos(θ) | Cosine of the angle |
| sec(θ) | Secant of the angle = 1/cos(θ) |
The secant function is one of the six standard trigonometric functions. It is the reciprocal of cosine, and together with cosine it satisfies the Pythagorean identity: sec²(θ) = 1 + tan²(θ). The secant function has a range of (−∞, −1] ∪ [1, +∞) — its absolute value is always at least 1.
Special Angle Values
| θ (degrees) | cos(θ) | sec(θ) |
|---|---|---|
| 0° | 1 | 1 |
| 30° | √3/2 | 2√3/3 |
| 45° | √2/2 | √2 |
| 60° | 1/2 | 2 |
| 90° | 0 | undefined |
Assumptions and Limits
The input can be any real number. Values near 90°, 270°, etc. (where cos approaches 0) produce very large secant values that may display as Infinity for exact multiples.
Use Cases for Secant Calculator
The Secant Calculator is useful in:
- Trigonometry learning — Understand the reciprocal relationship between secant and cosine.
- Calculus — Compute sec(θ) values needed for derivatives: d/dx[tan(x)] = sec²(x).
- Trigonometric identities — Verify identities such as sec²(θ) − tan²(θ) = 1.
- Engineering — Evaluate sec in structural analysis, beam deflection, or signal processing formulas.
- Homework checking — Confirm manually computed secant values against the calculator.
The Secant Calculator displays both cos(θ) and sec(θ) together, making the reciprocal relationship visually clear and reducing the risk of arithmetic errors when computing 1/cos(θ) by hand.