Triple Angle Formulas/Cosine

Theorem

$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$

where $\cos$ denotes cosine.

Example: $2 \cos 3 \theta + 1$

$2 \cos 3 \theta + 1 = \paren {\cos \theta - \cos \dfrac {2 \pi} 9} \paren {\cos \theta - \cos \dfrac {4 \pi} 9} \paren {\cos \theta - \cos \dfrac {8 \pi} 9}$

Proof 1

\(\ds \cos 3 \theta\)

\(=\)

\(\ds \cos \paren {2 \theta + \theta}\)

\(\ds \)

\(=\)

\(\ds \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta\)

Cosine of Sum

\(\ds \)

\(=\)

\(\ds \paren {\cos^2 \theta - \sin^2 \theta} \cos \theta - \paren {2 \sin \theta \cos \theta} \sin \theta\)

Double Angle Formula for Cosine and Double Angle Formula for Sine

\(\ds \)

\(=\)

\(\ds \cos^3 \theta - \sin^2 \theta \cos \theta - 2 \sin^2 \theta \cos \theta\)

\(\ds \)

\(=\)

\(\ds \cos^3 \theta - \paren {1 - \cos^2 \theta} \cos \theta - 2 \paren {1 - \cos^2 \theta} \cos \theta\)

Sum of Squares of Sine and Cosine

\(\ds \)

\(=\)

\(\ds \cos^3 \theta - \cos \theta + \cos^3 \theta - 2 \cos \theta + 2 \cos^3 \theta\)

multiplying out

\(\ds \)

\(=\)

\(\ds 4 \cos^3 \theta - 3 \cos \theta\)

gathering terms

$\blacksquare$

Proof 2

We have:

\(\ds \cos 3 \theta + i \sin 3 \theta\)

\(=\)

\(\ds \paren {\cos \theta + i \sin \theta}^3\)

De Moivre's Formula

\(\ds \)

\(=\)

\(\ds \paren {\cos \theta}^3 + \binom 3 1 \paren {\cos \theta}^2 \paren {i \sin \theta}\)

Binomial Theorem

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds \binom 3 2 \paren {\cos \theta} \paren {i \sin \theta}^2 + \paren {i \sin \theta}^3\)

\(\ds \)

\(=\)

\(\ds \cos^3 \theta + 3 i \cos^2 \theta \sin \theta + 3 i^2 \cos \theta \sin^2 \theta + i^3 \sin^3 \theta\)

substituting for binomial coefficients

\(\ds \)

\(=\)

\(\ds \cos^3 \theta + 3 i \cos^2 \theta \sin \theta - 3 \cos \theta \sin^2 \theta - i \sin^3 \theta\)

$i^2 = -1$

\(\text {(1)}: \quad\)

\(\ds \)

\(=\)

\(\ds \cos^3 \theta - 3 \cos \theta \sin^2 \theta\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds i \paren {3 \cos^2 \theta \sin \theta - \sin^3 \theta}\)

rearranging

Hence:

\(\ds \cos 3 \theta\)

\(=\)

\(\ds \cos^3 \theta - 3 \cos \theta \sin^2 \theta\)

equating real parts in $(1)$

\(\ds \)

\(=\)

\(\ds \cos^3 \theta - 3 \cos \theta \paren {1 - \cos^2 \theta}\)

Sum of Squares of Sine and Cosine

\(\ds \)

\(=\)

\(\ds 4 \cos^3 \theta - 3 \cos \theta\)

multiplying out and gathering terms

$\blacksquare$

Proof 3

\(\ds \cos 3 \theta\)

\(=\)

\(\ds \cos \paren {2 \theta + \theta}\)

\(\ds \)

\(=\)

\(\ds \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta\)

Cosine of Sum

\(\ds \)

\(=\)

\(\ds \paren {2 \cos^2 \theta - 1} \cos \theta - \paren {2 \sin \theta \cos \theta} \sin \theta\)

Double Angle Formula for Cosine and Double Angle Formula for Sine

\(\ds \)

\(=\)

\(\ds 2 \cos^3 \theta - \cos \theta - 2 \paren {1 - \cos^2 \theta} \cos \theta\)

Sum of Squares of Sine and Cosine after algebra

\(\ds \)

\(=\)

\(\ds 2 \cos^3 \theta - \cos \theta - 2 \cos \theta + 2 \cos^3 \theta\)

multiplying everything out

\(\ds \)

\(=\)

\(\ds 4 \cos^3 \theta - 3 \cos \theta\)

gathering terms

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.45$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $90 \ \text{(b)}$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): multiple-angle formulae
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): multiple-angle formulae