Sine of 45 Degrees/Proof 4
Jump to navigation
Jump to search
Theorem
- $\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$
Proof
\(\ds \sin 45 \degrees\) | \(=\) | \(\ds \map \sin {60 \degrees - 15 \degrees}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin 60 \degrees \cos 15 \degrees - \cos 60 \degrees \sin 15 \degrees\) | Sine of Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {\sqrt 3} 2} \paren {\frac {\sqrt 6 + \sqrt 2} 4} - \paren {\frac 1 2} \paren {\frac {\sqrt 6 - \sqrt 2} 4}\) | Sine of $60 \degrees$, Cosine of $15 \degrees$, Cosine of $60 \degrees$, Sine of $15 \degrees$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 8 \paren {\sqrt 3 \paren {\sqrt 6 + \sqrt 2} - \paren {\sqrt 6 - \sqrt 2} }\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 8 \paren {3 \sqrt 2 + \sqrt 6 - \sqrt 6 + \sqrt 2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 8 \paren {3 \sqrt 2 + \sqrt 2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 8 \paren {4 \sqrt 2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sqrt 2} 2\) |
$\blacksquare$