How to Use Trigonometric Identities Calculator
The Trigonometric Identities Calculator helps you verify common trigonometric identities for any angle instantly. Choose your preferred unit — degrees or radians — enter the angle, and the calculator checks all seven standard identities at once.
- Select angle unit — choose Degrees or Radians.
- Enter the angle θ — type any real number in the input field.
- Review the results — the Trigonometric Identities Calculator shows each identity, its left-hand side value, right-hand side value, and whether it holds.
The basic trig values (sin θ, cos θ, tan θ) are displayed alongside the identity table so you can cross-reference your inputs easily.
Formula & Theory - Trigonometric Identities Calculator
The Trigonometric Identities Calculator verifies the following key identities:
Pythagorean identities:
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
Double-angle formulas:
sin(2θ) = 2·sinθ·cosθ
cos(2θ) = cos²θ − sin²θ
Half-angle formulas:
sin²(θ/2) = (1 − cosθ) / 2
cos²(θ/2) = (1 + cosθ) / 2| Symbol | Meaning |
|---|---|
| θ | The input angle |
| sin, cos, tan | Sine, cosine, tangent |
| sec, csc, cot | Secant, cosecant, cotangent |
The identities are derived from the unit circle definition of trigonometric functions. The Pythagorean identities follow directly from x² + y² = r² applied to a point on the unit circle. Double- and half-angle formulas are derived via angle addition rules.
Assumptions and Limits
The calculator works with real number angles only. Identities involving tan, cot, sec, or csc are marked as “Undefined” at angles where those functions have poles. The tool is intended for educational purposes and numerical verification; it does not perform symbolic algebra.
Use Cases for Trigonometric Identities Calculator
The Trigonometric Identities Calculator is useful whenever you need to quickly confirm a trigonometric relationship. Common uses include:
- Homework checking — Verify that a simplified expression holds for a specific angle before submitting an assignment.
- Formula revision — Quickly recall which half-angle or double-angle identity applies to a given angle.
- Tutoring and teaching — Demonstrate why identities hold numerically to students who are new to trigonometry.
- Engineering spot-checks — Confirm that a computed trigonometric expression matches the expected identity in a numerical simulation.
The Trigonometric Identities Calculator displays both the LHS and RHS values, making it easy to spot discrepancies and understand how close a numerical result is to the theoretical identity.