Triple Angle Calculator

Compute sin(3θ), cos(3θ), tan(3θ), cot(3θ), sec(3θ), csc(3θ) for any angle. Visualize on the unit circle and explore triple‑angle identities.

0° 30° (π/6) 45° (π/4) 60° (π/3) 90° (π/2) 120° (2π/3) 180° (π)
Triple Angle Values
sin(3θ) 1.000000
cos(3θ) 0.000000
tan(3θ) undefined
cot(3θ) 0.000000
sec(3θ) undefined
csc(3θ) 1.000000
θ (degrees) 30.00°
θ (radians) 0.523599
3θ (degrees) 90.00°
3θ (radians) 1.570796

Triple‑angle identities:

  • sin(3θ) = 3 sin θ – 4 sin³ θ
  • cos(3θ) = 4 cos³ θ – 3 cos θ
  • tan(3θ) = (3 tan θ – tan³ θ) / (1 – 3 tan² θ)
  • cot(3θ) = (cot³ θ – 3 cot θ) / (3 cot² θ – 1)
  • sec(3θ) = 1 / cos(3θ)
  • csc(3θ) = 1 / sin(3θ)

Understanding Triple‑Angle Formulas

Triple‑angle formulas express trigonometric functions of three times an angle in terms of powers of the original angle. They are derived from the sum formulas and double‑angle identities, and are essential in advanced trigonometry, calculus, and physics (e.g., triple‑angle harmonics, pendulum motion).

Derivation from Sum and Double‑Angle Formulas

Using sin(α+β) and cos(α+β) with α = 2θ, β = θ:

  • sin(3θ) = sin(2θ + θ) = sin 2θ cos θ + cos 2θ sin θ
  • cos(3θ) = cos(2θ + θ) = cos 2θ cos θ – sin 2θ sin θ

Substituting sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ – sin²θ (or its other forms) and simplifying yields the formulas above. For example:

sin(3θ) = 3 sin θ – 4 sin³ θ
cos(3θ) = 4 cos³ θ – 3 cos θ

These forms are known as the “triple‑angle identities” and are useful for solving cubic equations and analyzing periodic functions.

Geometric Interpretation

On the unit circle, the angle 3θ represents three rotations of θ. The point (cos 3θ, sin 3θ) can be expressed in terms of (cos θ, sin θ) using the identities above. The visualization at the top shows the radii for θ (blue) and 3θ (red).

Common Applications

  • Solving cubic equations: The triple‑angle formulas link trigonometric functions to cubic polynomials (e.g., 4 cos³θ – 3 cos θ = cos 3θ).
  • Physics: Triple‑angle harmonics appear in wave superposition and nonlinear oscillations.
  • Signal processing: Triple‑angle modulation schemes.
  • Geometry: Calculating coordinates of points after threefold rotation.
  • Calculus: Integration of powers of sine/cosine can be reduced using triple‑angle forms.
  • Number theory: Chebyshev polynomials are closely related.

Related Identities

  • Double‑angle formulas: sin(2θ), cos(2θ), etc. – the foundation for triple‑angle derivations.
  • Power‑reduction formulas: Express sin²θ, cos²θ, sin³θ, cos³θ in terms of multiple angles.
  • Multiple‑angle formulas (general): sin(nθ) and cos(nθ) expressed via Chebyshev polynomials.

Tip: When tan(3θ) or sec(3θ) is undefined, check whether cos(3θ) = 0. Similarly, csc and cot are undefined when sin(3θ) = 0. The calculator will display “undefined” in those cases.