Triple Angle Formulas/Cosine/Proof 3

Theorem

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

Proof

\(\ds \cos 3 \theta\)

\(=\)

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

\(\ds \)

\(=\)

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

Cosine of Sum

\(\ds \)

\(=\)

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

Double Angle Formula for Cosine and Double Angle Formula for Sine

\(\ds \)

\(=\)

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

Sum of Squares of Sine and Cosine

\(\ds \)

\(=\)

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

gathering terms

$\blacksquare$

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid