find the following.\ntan \\frac{\\theta}{2}, given \\sin \\theta=-\\frac{12}{13}, 270^{\\circ}<\\theta<360^{\…

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})