Question

find the following.
tan \frac{\theta}{2}, given \sin \theta=-\frac{12}{13}, 270^{\circ}<\theta<360^{\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 \(\cos\theta\)

Given \(\sin\theta =-\frac{12}{13}\) and \(270^{\circ}<\theta < 360^{\circ}\).
Using the identity \(\sin^{2}\theta+\cos^{2}\theta = 1\), we have \(\cos^{2}\theta=1-\sin^{2}\theta\).
Substitute \(\sin\theta =-\frac{12}{13}\): \(\cos^{2}\theta=1 - (-\frac{12}{13})^{2}=1-\frac{144}{169}=\frac{169 - 144}{169}=\frac{25}{169}\).
Since \(\theta\) is in the fourth - quadrant (\(270^{\circ}<\theta < 360^{\circ}\)), \(\cos\theta>0\), 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:
\(\tan\frac{\theta}{2}=\frac{-\frac{12}{13}}{1+\frac{5}{13}}\).
First, simplify the denominator: \(1+\frac{5}{13}=\frac{13 + 5}{13}=\frac{18}{13}\).
Then, \(\tan\frac{\theta}{2}=\frac{-\frac{12}{13}}{\frac{18}{13}}\).
When dividing by a fraction, we multiply by its reciprocal: \(\tan\frac{\theta}{2}=-\frac{12}{13}\times\frac{13}{18}\).
Cancel out the common factor of \(13\): \(\tan\frac{\theta}{2}=-\frac{12}{18}\).
Simplify the fraction: \(\tan\frac{\theta}{2}=-\frac{2}{3}\).

Answer:

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