Working with an angle identity that involves tripling? This triple angle formula calculator computes all six trigonometric functions for 3θ given any input angle θ.
Triple Angle Calculator
Using the triple angle calculator
Enter your angle in the Angle (θ) field and select Degrees or Radians. The calculator instantly shows all six trigonometric functions for triple that angle: sine, cosine, tangent, and their reciprocals (cosecant, secant, cotangent).
The result title displays the computed angle – enter 20° and you'll see results for 60°.
What are the triple angle formulas?
The triple angle formulas express trigonometric functions of 3θ using only functions of θ. They're more complex than the double angle formulas but follow the same derivation pattern.
Triple angle formula for sine
Triple angle formula for cosine
Triple angle formula for tangent
Example: triple angle of 20°
Let's compute sin(60°) using the triple angle formula with θ = 20°. First, sin(20°) ≈ 0.342:
This matches sin(60°) = √3/2 ≈ 0.866.
Deriving the triple angle formulas
The triple angle formulas come from applying the angle addition formula twice. Start with:
Using the angle addition formula:
Substitute the double angle formulas:
Replace cos²θ with 1 - sin²θ:
The cubic connection
Notice something interesting: the triple angle formulas are cubic polynomials. This connects to the classical problem of trisecting an angle – dividing an angle into three equal parts using only compass and straightedge.
The ancient Greeks tried for centuries to solve this. In 1837, Pierre Wantzel proved it's impossible in general. The proof relies on the fact that the triple angle formula leads to a cubic equation, and most cubic equations can't be solved with compass and straightedge constructions (which only allow square roots).
So when you enter 60° and get results for 20°, you're essentially "trisecting" 60° – something the Greeks couldn't do with their tools, but our calculator handles instantly.
When to use triple angle formulas
Triple angle formulas appear less frequently than half or double angle formulas, but they're useful for:
- Solving cubic trig equations: Equations like
sin(3θ) = kcan be converted to cubic polynomials - Finding exact values: Compute
sin(15°)viasin(45°) = sin(3×15°) - Fourier analysis: Expressing higher harmonics in terms of fundamental frequencies
- Competition math: Trig identity problems often involve triple (or higher) angle formulas
