Question

if $\theta=\frac{7\pi}{6}$, then find exact values for the following:
$\sec(\theta)$ equals
$\csc(\theta)$ equals
$\tan(\theta)$ equals
$\cot(\theta)$ equals

Explanation:

Step1: Determine the reference angle

The angle \(\theta=\frac{7\pi}{6}\) is in the third - quadrant. The reference angle \(\theta_{r}=\frac{7\pi}{6}-\pi=\frac{\pi}{6}\)

Step2: Find \(\cos\theta\) and \(\sec\theta\)

We know that \(\cos\theta =-\cos\theta_{r}\) (since \(\cos\) is negative in the third - quadrant). Since \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\), then \(\cos\frac{7\pi}{6}=-\frac{\sqrt{3}}{2}\)
\(\sec\theta=\frac{1}{\cos\theta}\), so \(\sec\frac{7\pi}{6}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2\sqrt{3}}{3}\)

Step3: Find \(\sin\theta\) and \(\csc\theta\)

We know that \(\sin\theta =-\sin\theta_{r}\) (since \(\sin\) is negative in the third - quadrant). Since \(\sin\frac{\pi}{6}=\frac{1}{2}\), then \(\sin\frac{7\pi}{6}=-\frac{1}{2}\)
\(\csc\theta=\frac{1}{\sin\theta}\), so \(\csc\frac{7\pi}{6}=\frac{1}{-\frac{1}{2}}=-2\)

Step4: Find \(\tan\theta\)

We know that \(\tan\theta=\frac{\sin\theta}{\cos\theta}\). \(\sin\frac{7\pi}{6}=-\frac{1}{2}\) and \(\cos\frac{7\pi}{6}=-\frac{\sqrt{3}}{2}\)
\(\tan\frac{7\pi}{6}=\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\)

Step5: Find \(\cot\theta\)

Since \(\cot\theta=\frac{1}{\tan\theta}\), then \(\cot\frac{7\pi}{6}=\sqrt{3}\)

Answer:

\(\sec(\frac{7\pi}{6})=-\frac{2\sqrt{3}}{3}\)
\(\csc(\frac{7\pi}{6})=-2\)
\(\tan(\frac{7\pi}{6})=\frac{\sqrt{3}}{3}\)
\(\cot(\frac{7\pi}{6})=\sqrt{3}\)