# How do you determine costheta given sintheta=1/4,0<theta<pi/2?

Dec 3, 2016

Use the identity $\cos \left(\theta\right) = \pm \sqrt{1 - {\sin}^{2} \left(\theta\right)}$

#### Explanation:

Use the identity $\cos \left(\theta\right) = \pm \sqrt{1 - {\sin}^{2} \left(\theta\right)}$

Given that $0 < \theta < \frac{\pi}{2}$ then the cosine function is must be positive and the we must make the $\pm$ be only positive:

$\cos \left(\theta\right) = \sqrt{1 - {\sin}^{2} \left(\theta\right)}$

Substitute ${\left(\frac{1}{4}\right)}^{2}$ for ${\sin}^{2} \left(\theta\right)$:

$\cos \left(\theta\right) = \sqrt{1 - {\left(\frac{1}{4}\right)}^{2}}$

$\cos \left(\theta\right) = \sqrt{1 - \frac{1}{16}}$

$\cos \left(\theta\right) = \sqrt{\frac{15}{16}}$

$\cos \left(\theta\right) = \frac{\sqrt{15}}{4}$