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

 $\displaystyle \cos 3 \theta$ $=$ $\displaystyle \cos \paren {2 \theta + \theta}$ $\displaystyle$ $=$ $\displaystyle \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta$ Cosine of Sum $\displaystyle$ $=$ $\displaystyle \paren {\cos^2 \theta - \sin^2 \theta} \cos \theta - \paren {2 \sin \theta \cos \theta} \sin \theta$ Double Angle Formula for Cosine and Double Angle Formula for Sine $\displaystyle$ $=$ $\displaystyle \cos^3 \theta - \sin^2 \theta \cos \theta - 2 \sin^2 \theta \cos \theta$ $\displaystyle$ $=$ $\displaystyle \cos^3 \theta - \paren {1 - \cos^2 \theta} \cos \theta - 2 \paren {1 - \cos^2 \theta} \cos \theta$ Sum of Squares of Sine and Cosine $\displaystyle$ $=$ $\displaystyle \cos^3 \theta - \cos \theta + \cos^3 \theta - 2 \cos \theta + 2 \cos^3 \theta$ multiplying out $\displaystyle$ $=$ $\displaystyle 4 \cos^3 \theta - 3 \cos \theta$ gathering terms

$\blacksquare$

Proof 2

We have:

 $\displaystyle \cos 3 \theta + i \sin 3 \theta$ $=$ $\displaystyle \paren {\cos \theta + i \sin \theta}^3$ De Moivre's Formula $\displaystyle$ $=$ $\displaystyle \paren {\cos \theta}^3 + \binom 3 1 \paren {\cos \theta}^2 \paren {i \sin \theta}$ Binomial Theorem $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \binom 3 2 \paren {\cos \theta} \paren {i \sin \theta}^2 + \paren {i \sin \theta}^3$ $\displaystyle$ $=$ $\displaystyle \cos^3 \theta + 3 i \cos^2 \theta \sin \theta + 3 i^2 \cos \theta \sin^2 \theta + i^3 \sin^3 \theta$ substituting for binomial coefficients $\displaystyle$ $=$ $\displaystyle \cos^3 \theta + 3 i \cos^2 \theta \sin \theta - 3 \cos \theta \sin^2 \theta - i \sin^3 \theta$ $i^2 = -1$ $\text {(1)}: \quad$ $\displaystyle$ $=$ $\displaystyle \cos^3 \theta - 3 \cos \theta \sin^2 \theta$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle i \paren {3 \cos^2 \theta \sin \theta - \sin^3 \theta}$ rearranging

Hence:

 $\displaystyle \cos 3 \theta$ $=$ $\displaystyle \cos^3 \theta - 3 \cos \theta \sin^2 \theta$ equating real parts in $(1)$ $\displaystyle$ $=$ $\displaystyle \cos^3 \theta - 3 \cos \theta \paren {1 - \cos^2 \theta}$ Sum of Squares of Sine and Cosine $\displaystyle$ $=$ $\displaystyle 4 \cos^3 \theta - 3 \cos \theta$ multiplying out and gathering terms

$\blacksquare$

Proof 3

 $\displaystyle \cos 3 \theta$ $=$ $\displaystyle \cos \paren {2 \theta + \theta}$ $\displaystyle$ $=$ $\displaystyle \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta$ Cosine of Sum $\displaystyle$ $=$ $\displaystyle \paren {2 \cos^2 \theta - 1} \cos \theta - 2 \sin^2 \theta \cos \theta$ Double Angle Formula for Cosine and Double Angle Formula for Sine $\displaystyle$ $=$ $\displaystyle \paren {2 \cos^2 \theta - 1 - 2 \paren {1 - 2 \cos^2 \theta} } \cos \theta$ Sum of Squares of Sine and Cosine $\displaystyle$ $=$ $\displaystyle 4 \cos^3 \theta - 3 \cos \theta$ gathering terms

$\blacksquare$