Question

given that \\( \cos ( \theta ) = - \frac { 1 2 } { 1 3 } \\), what is the numerical value of \\( \cot ( \theta ) \\), if \\( \pi < \theta < \frac { 3 \pi } { 2 } \\)?
\\( \frac { 1 3 } { 1 2 } \\)
\\( \frac { 1 2 } { 5 } \\)
\\( - \frac { 5 } { 1 2 } \\)
\\( - \frac { 1 3 } { 1 2 } \\)

Explanation:

Step1: Find $\sin(\theta)$

Use the identity $\sin^{2}\theta+\cos^{2}\theta = 1$.
Given $\cos\theta=-\frac{12}{13}$, then $\sin^{2}\theta=1-\cos^{2}\theta=1 - (-\frac{12}{13})^{2}=1-\frac{144}{169}=\frac{169 - 144}{169}=\frac{25}{169}$.
Since $\pi<\theta<\frac{3\pi}{2}$ (third - quadrant where $\sin\theta<0$), so $\sin\theta=-\frac{5}{13}$.

Step2: Calculate $\cot\theta$

Use the formula $\cot\theta=\frac{\cos\theta}{\sin\theta}$.
Substitute $\cos\theta =-\frac{12}{13}$ and $\sin\theta=-\frac{5}{13}$ into the formula: $\cot\theta=\frac{-\frac{12}{13}}{-\frac{5}{13}}=\frac{12}{5}$.

Answer:

$\frac{12}{5}$