Power Reduction Formulas/Sine Cubed

From ProofWiki
Jump to navigation Jump to search

Theorem

$\sin^3 x = \dfrac {3 \sin x - \sin 3 x} 4$

where $\sin$ denotes sine.


Proof 1

\(\displaystyle \sin 3 x\) \(=\) \(\displaystyle 3 \sin x - 4 \sin^3 x\) Triple Angle Formula for Sine
\(\displaystyle \implies \ \ \) \(\displaystyle 4 \sin^3 x\) \(=\) \(\displaystyle 3 \sin x - \sin 3 x\) rearranging
\(\displaystyle \implies \ \ \) \(\displaystyle \sin^3 x\) \(=\) \(\displaystyle \frac {3 \sin x - \sin 3 x} 4\) dividing both sides by $4$

$\blacksquare$


Proof 2

\(\displaystyle \sin^3 x\) \(=\) \(\displaystyle \paren {\frac {e^{i x} - e^{-i x} } {2 i} }^3\) Sine Exponential Formulation
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {e^{i x} - e^{-i x} }^3} {8 i^3}\) rearranging
\(\displaystyle \) \(=\) \(\displaystyle -\frac 1 {8 i} \paren {\paren {e^{i x} }^3 - 3 \paren {e^{i x} }^2 \paren {e^{-i x} } + 3 \paren {e^{i x} } \paren {e^{-i x} }^2 - \paren {e^{-i x} }^3}\) multiplying out
\(\displaystyle \) \(=\) \(\displaystyle -\frac 1 {8 i} \paren {e^{3 i x} - 3 e^{i x} + 3 e^{-i x} - e^{-3 i x} }\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac 3 4 \paren {\frac {e^{i x} - e^{-i x} } {2 i} } - \frac 1 4 \paren {\frac {e^{3 i x} - e^{-3 i x} } {2 i} }\) gathering terms
\(\displaystyle \) \(=\) \(\displaystyle \frac {3 \sin x - \sin 3 x} 4\) Sine Exponential Formulation

$\blacksquare$


Sources