Question

1. \\( \tan 157.5 ^ { circ } \times 2 = \\)

\\( \frac { sin 315 } { 1 + cos 315 } = \frac { - \frac { sqrt { 2 } } { 2 } } { 1 + \frac { sqrt { 2 } } { 2 } } = \frac { - \frac { sqrt { 2 } } { 2 } } { \frac { 2 + \sqrt { 2 } } { 2 } } = \frac { \sqrt { 2 } } { 2 } \cdot \frac { 2 } { 2 + \sqrt { 2 } } \\)
\\( \frac { \sqrt { 2 } } { 2 + \sqrt { 2 } } \frac { 2 - \sqrt { 2 } } { 2 - \sqrt { 2 } } \\)
\\( = \frac { 2 \sqrt { 2 } - 2 } { 2 } = 1 - \sqrt { 2 } \\)

1. \\( sin \frac { 11 \pi } { 12 } \\)

\\( \frac { \sqrt { 2 - \sqrt { 3 } } } { 4 } \\)
\\( \frac { \sqrt { 2 - \sqrt { 3 } } } { 2 } \\)

1. if \\( cos \theta = - \frac { 3 } { 8 } \\) and \\( \frac { \pi } { 2 } < \theta < \pi \\), find \\( sin \frac { \theta } { 2 } \\).

\\( sin \frac { \theta } { 2 } = \pm \sqrt { \frac { 1 - cos \theta } { 2 } } \\)
\\( = \sqrt { \frac { 1 + 3 } { 8 } } \sqrt { \frac { 11 } { 8 } \cdot \frac { 1 } { 2 } } \\)

1. if \\( sin \theta = - \frac { 12 } { 13 } \\) and \\( \frac { 3 \pi } { 2 } < \theta < 2 \pi \\), find \\( \tan \frac { \theta } { 2 } \\).
2. if \\( \tan \theta = 4 \sqrt { 3 } \\) and \\( \pi < \theta < \frac { 3 \pi } { 2 } \\), find \\( cos \frac { \theta } { 2 } \\).

Explanation:

Step1: Determine the quadrant of \(\frac{\theta}{2}\)

Given \(\frac{\pi}{2}<\theta<\pi\), then \(\frac{\pi}{4}<\frac{\theta}{2}<\frac{\pi}{2}\). So \(\sin\frac{\theta}{2}>0\)

Step2: Use the half - angle formula

The half - angle formula for \(\sin\frac{\alpha}{2}\) is \(\sin\frac{\alpha}{2}=\sqrt{\frac{1 - \cos\alpha}{2}}\)
Substitute \(\alpha=\theta\) and \(\cos\theta =-\frac{3}{8}\) into the formula:

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Answer:

\(\frac{\sqrt{11}}{4}\)