# If sin θ = 1/2 and angle θ is acute, how do you find the other five ratios?

Apr 2, 2018

color(green)(cos theta = sqrt3 / 2, sec theta = 2 / sqrt3, csc theta = 2, tan theta = 1 / sqrt3, cot theta = sqrt3

#### Explanation:

#"Given " sin theta = 1/2, " and the angle is acute or less than " 90^2 " which lies in the first quadrant.

This makes all the trigonometric functions positive.

${\cos}^{2} \theta = 1 - {\sin}^{2} \theta = 1 - \frac{1}{4} = \frac{3}{4}$

Hence, $\cos \theta = \frac{\sqrt{3}}{2}$

$\tan \theta = \sin \frac{\theta}{\cos} \theta = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}$

$\cot \theta = \frac{1}{\tan} \theta = 1 \left(\frac{1}{\sqrt{3}}\right) = \sqrt{3}$

$\csc \theta = \frac{1}{\sin} \theta = 1 \left(\frac{1}{2}\right) = 2$

$\sec \theta = \frac{1}{\cos} \theta = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}$