Triple Angle Formula Calculator (sin 3θ, cos 3θ)

Three times the angle, in terms of the original

The triple angle identities rewrite a trig function of 3θ using only functions of θ. They extend the double angle formulas one step further:

sin(3θ) = 3 sin θ − 4 sin³θ cos(3θ) = 4 cos³θ − 3 cos θ tan(3θ) = (3 tan θ − tan³θ) / (1 − 3 tan²θ)

You derive them by writing 3θ as 2θ + θ and applying the angle addition formula, then substituting the double angle identities. This calculator takes an angle and returns all three, next to a direct check against sin(3θ) computed the ordinary way.

Notice the clean structure

The sine formula uses only powers of sine, the cosine formula only powers of cosine. That symmetry makes them easier to remember than they first look. The tangent version follows from dividing the other two, and it blows up wherever 1 − 3 tan²θ hits zero.

The angle-trisection connection

The cosine identity hides one of the most famous results in mathematics. Rearranged, 4cos³θ − 3cosθ = cos(3θ) is a cubic in cos θ. Trisecting a general angle would mean solving that cubic with only compass and straightedge, and Pierre Wantzel proved in 1837 that this is impossible for most angles. So this tidy-looking identity is the reason one of the three classical Greek construction problems has no solution.

Worked example

Set θ = 30°. Then sin(3 × 30°) = sin(90°) should be 1. The formula gives 3 sin30° − 4 sin³30° = 3(0.5) − 4(0.125) = 1.5 − 0.5 = 1, exactly as expected.

Where they show up

Third-harmonic distortion in audio and power systems, where a device pushes energy to three times the input frequency. Trigonometric solutions of cubic equations. Wave interference and some orbital-mechanics work. They turn up less often than the double angle formulas, but reliably whenever a frequency or an angle gets tripled.