Triple Angle Formulas
Triple angle formulas for sin(3x), cos(3x), and tan(3x).
Derived step-by-step with worked examples and applications.
The Formula
cos(3θ) = 4cos3θ − 3cosθ
tan(3θ) = (3tanθ − tan3θ) / (1 − 3tan2θ)
The triple angle formulas express trigonometric functions of three times an angle in terms of functions of the original angle. They are natural extensions of the double angle formulas. To derive sin(3θ), you write it as sin(2θ + θ) and apply the angle addition formula, then substitute the double angle identities. The same approach works for cosine and tangent.
These formulas are particularly useful in advanced trigonometry, signal processing, and physics. In signal processing, triple angle formulas help analyze harmonic distortion where a system produces output at three times the input frequency. In physics, they appear in the study of orbital mechanics and wave interference patterns.
The cosine triple angle formula has an elegant connection to the famous problem of trisecting an angle. The equation 4cos³θ − 3cosθ = cos(3θ) can be rearranged into a cubic equation that, in general, cannot be solved using compass and straightedge alone. This is why angle trisection was proven impossible as a general construction in classical geometry.
Notice the symmetric structure: the sine formula involves only powers of sine, and the cosine formula involves only powers of cosine. The tangent formula follows from dividing sine by cosine. These patterns make the formulas easier to memorize and apply.
Variables
| Symbol | Meaning |
|---|---|
| θ | The original angle (in radians or degrees) |
| sin, cos, tan | Standard trigonometric functions |
Example 1
Find sin(90°) using the triple angle formula with θ = 30°.
sin(3 × 30°) = 3sin(30°) − 4sin³(30°)
= 3(0.5) − 4(0.5)³ = 1.5 − 4(0.125) = 1.5 − 0.5
sin(90°) = 1 ✔
Example 2
Find cos(60°) using the triple angle formula with θ = 20°.
cos(3 × 20°) = 4cos³(20°) − 3cos(20°)
cos(20°) ≈ 0.9397
= 4(0.9397)³ − 3(0.9397) = 4(0.8298) − 2.8191 = 3.3192 − 2.8191
cos(60°) ≈ 0.5 ✔
When to Use It
Triple angle formulas come up in various mathematical and scientific contexts.
- Simplifying trigonometric expressions involving triple angles
- Signal processing: analyzing third-harmonic distortion
- Solving cubic equations through trigonometric substitution
- Physics: wave interference and orbital mechanics calculations
- Computer graphics: generating smooth curves and rotations