Double-angle and Half-angle Formulas
Examples:
1. Find the exact value of \(\sin \frac{\pi}{8} \).
SOLUTION
2. Find the exact value of \(\sin 15^\circ\).
SOLUTION
3. Find the exact value of \( \cos \left(2
\sin^{-1}\frac{3}{5}\right)\).
SOLUTION
4. If \( \cos \alpha = \frac{5}{13},
\frac{3\pi}{2}< \alpha < 2\pi \). Find the exact value of \( \tan
(2\alpha)\).
SOLUTION
5. If If \(\sin \theta = -\frac{\sqrt 3}{3},
\frac{3\pi}{2}< \theta < 2\pi \). Find the exact value of \( \cos
\frac{\theta}{2}\).
SOLUTION
Establish each identity.
1. \(\displaystyle{\frac{\cot \theta - \tan \theta}{\cot \theta +
\tan \theta} = \cos (2\theta)}\)
SOLUTION
2. \(\displaystyle{\frac{\cos \theta + \sin \theta}{\cos \theta -
\sin \theta} - \frac{\cos \theta - \sin \theta}{\cos \theta + \sin
\theta} = 2\tan (2\theta)}\)
SOLUTION
3. \(\displaystyle{\frac{\cos (2\theta)}{1 + \sin (2\theta)} =
\frac{\cot \theta - 1}{\cot\theta + 1}}\)
SOLUTION
4. \(\displaystyle{\cos^4 \theta - \sin^4 \theta = \cos(2\theta)}\)
SOLUTION
5. \(\displaystyle{\cot(2\theta) = \frac{1}{2}(\cot \theta - \tan
\theta)}\) SOLUTION