Power Reduction Formulas/Sine Cubed
Jump to navigation
Jump to search
Theorem
- $\sin^3 x = \dfrac {3 \sin x - \sin 3 x} 4$
where $\sin$ denotes sine.
Proof 1
\(\ds \sin 3 x\) | \(=\) | \(\ds 3 \sin x - 4 \sin^3 x\) | Triple Angle Formula for Sine | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 4 \sin^3 x\) | \(=\) | \(\ds 3 \sin x - \sin 3 x\) | rearranging | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin^3 x\) | \(=\) | \(\ds \frac {3 \sin x - \sin 3 x} 4\) | dividing both sides by $4$ |
$\blacksquare$
Proof 2
\(\ds \sin^3 x\) | \(=\) | \(\ds \paren {\frac {e^{i x} - e^{-i x} } {2 i} }^3\) | Euler's Sine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {e^{i x} - e^{-i x} }^3} {8 i^3}\) | rearranging | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 {8 i} \paren {\paren {e^{i x} }^3 - 3 \paren {e^{i x} }^2 \paren {e^{-i x} } + 3 \paren {e^{i x} } \paren {e^{-i x} }^2 - \paren {e^{-i x} }^3}\) | multiplying out | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 {8 i} \paren {e^{3 i x} - 3 e^{i x} + 3 e^{-i x} - e^{-3 i x} }\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 3 4 \paren {\frac {e^{i x} - e^{-i x} } {2 i} } - \frac 1 4 \paren {\frac {e^{3 i x} - e^{-3 i x} } {2 i} }\) | gathering terms | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \sin x - \sin 3 x} 4\) | Euler's Sine Identity |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.55$