Answer
$\sin\theta$ = $\frac{\sqrt 3}{2}$
$\cos\theta$ = $\frac{1}{2}$
$\tan\theta$ = $\sqrt 3$
$\csc\theta$ = $\frac{2}{\sqrt 3}$
$\cot\theta$ = $\frac{1}{\sqrt 3}$
Work Step by Step
Given $\sec\theta$ = 2
By reciprocal identity-
$\cos\theta$ = $\frac{1}{\sec\theta}$ = $\frac{1}{2}$
We know from first Pythagorean identity that-
$\sin\theta$ = ± $\sqrt (1-\cos^{2}\theta)$
Given $\sec\theta$ is positive and $\sin\theta$ is also positive, therefore $\theta$ terminates in Q I.
OR
$\sin\theta$ = $\sqrt (1-\cos^{2}\theta)$
substitute the given value of $\cos\theta$-
$\sin\theta$= $\sqrt (1-(\frac{1}{2})^{2})$
$\sin\theta$ = $\sqrt (1-\frac{1}{4})$
$\sin\theta$ = $\sqrt (\frac{4-1}{4})$ = $\sqrt (\frac{3}{4})$
$\sin\theta$ = $\frac{\sqrt 3}{2}$
By ratio identity-
$\tan\theta$ = $\frac{\sin\theta}{\cos\theta}$
Substituting the values of $\sin\theta$ and $\cos\theta$-
$\tan\theta$ = $\frac{\sqrt 3/2}{ 1/2}$ = $\sqrt 3$
From reciprocal identities-
$\csc\theta$ = $\frac{1}{\sin\theta}$ = $\frac{1}{\sqrt 3/2}$ = $\frac{2}{\sqrt 3}$
$\cot\theta$ = $\frac{1}{\tan\theta}$ = $\frac{1}{\sqrt 3}$
