Triple Angle Formulae: sin 3θ, cos 3θ, tan 3θ Derivations

The triple-angle formulae express sin 3θ, cos 3θ and tan 3θ in terms of θ alone: sin 3θ = 3 sin θ − 4 sin³θ, cos 3θ = 4 cos³θ − 3 cos θ, and tan 3θ = (3 tan θ − tan³θ)/(1 − 3 tan²θ). They are derived by writing 3θ as 2θ + θ and applying the sum and double-angle formulae.

They matter because they connect trigonometry to cubic equations: solving sin 3θ = k or finding exact values like sin 18° passes through these cubic relations. They are also the first members of the Chebyshev polynomial family used in approximation theory.

Key Concepts

• Derivation idea: sin 3θ = sin(2θ + θ) = sin 2θ cos θ + cos 2θ sin θ; substituting the double-angle forms and simplifying gives 3 sin θ − 4 sin³θ.
• Cubic nature: each formula is cubic in the single-angle ratio, which is why triple angles lead to cubic equations.
• Cosine mirrors sine: cos 3θ = 4 cos³θ − 3 cos θ has the coefficients swapped compared to the sine version.
• Not three times the value: sin 3θ is emphatically not 3 sin θ; the −4 sin³θ correction keeps the result within [−1, 1].

Formulas

• Sine: sin 3θ = 3 sin θ − 4 sin³θ.
• Cosine: cos 3θ = 4 cos³θ − 3 cos θ.
• Tangent: tan 3θ = (3 tan θ − tan³θ)/(1 − 3 tan²θ), valid where tan θ is defined and 3 tan²θ ≠ 1.
• Assumptions: θ is any real angle; for the tangent form, θ must avoid odd multiples of 90° and the zeros of the denominator.

Worked Example

Verify the cosine triple-angle formula for θ = 60°.

1. Directly, 3θ = 180°, and cos 180° = −1.
2. From the formula with cos 60° = 1/2: cos 3θ = 4 cos³θ − 3 cos θ = 4(1/2)³ − 3(1/2).
3. Compute: 4(1/8) − 3/2 = 1/2 − 3/2 = −2/2 = −1.
4. The formula's result −1 matches cos 180° = −1 exactly.
5. Bonus check on sine: sin 180° = 0, and 3 sin 60° − 4 sin³60° = 3(√3/2) − 4(3√3/8) = (3√3/2) − (3√3/2) = 0. Both formulae confirmed.

Common Misconceptions

• sin 3θ ≠ 3 sin θ. Test θ = 30°: sin 90° = 1 but 3 sin 30° = 3/2, which is impossible for a sine value.
• Do not drop the cubic term. The −4 sin³θ is not a small correction; without it the formula is completely wrong.
• Sine and cosine versions differ. Mixing up 3 sin θ − 4 sin³θ with 4 cos³θ − 3 cos θ gives sign and coefficient errors.

Real-World Applications

• Exact values: deriving sin 18° and cos 36° by solving the cubic equations these formulae produce.
• Solving trig equations: equations like sin 3x = 1/2 become cubics in sin x.
• Approximation theory: cos 3θ = 4 cos³θ − 3 cos θ is the Chebyshev polynomial T₃, used in numerical methods.
• Signal processing: third harmonics in distorted waveforms follow the triple-angle relations.

The triple-angle formulae — sin 3θ = 3 sin θ − 4 sin³θ, cos 3θ = 4 cos³θ − 3 cos θ, and the tangent version — come from expanding sin(2θ + θ). Keep the cubic terms, never confuse them with simple multiplication by 3, and test with known angles like 60° to verify.