Question

find the following.

\\( \tan \frac { \theta } { 2 } \\), given \\( \sin \theta = \frac { 12 } { 13 } \\), \\( 90 ^ { \circ } < \theta < 180 ^ { \circ } \\)

\\( \tan \frac { \theta } { 2 } = \square \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find $\cos\theta$

Using the identity $\sin^{2}\theta+\cos^{2}\theta = 1$.
Given $\sin\theta=\frac{12}{13}$, then $\cos^{2}\theta=1 - (\frac{12}{13})^{2}=1-\frac{144}{169}=\frac{169 - 144}{169}=\frac{25}{169}$.
Since $90^{\circ}<\theta<180^{\circ}$, $\cos\theta<0$, so $\cos\theta=-\frac{5}{13}$.

Step2: Use the half - angle formula for tangent

The half - angle formula for tangent is $\tan\frac{\theta}{2}=\frac{\sin\theta}{1 + \cos\theta}$.
Substitute $\sin\theta=\frac{12}{13}$ and $\cos\theta=-\frac{5}{13}$ into the formula:
$\tan\frac{\theta}{2}=\frac{\frac{12}{13}}{1+( -\frac{5}{13})}=\frac{\frac{12}{13}}{\frac{13 - 5}{13}}=\frac{\frac{12}{13}}{\frac{8}{13}}$.

Answer:

$\frac{12}{8}=\frac{3}{2}$