Find the cosine of $ heta $ if $ sin( heta) = frac{1}{2} $ and $ heta $ is in the first quadrant

Given $ \sin(\theta) = \frac{1}{2} $ and $ \theta $ is in the first quadrant.

We know that $ \sin^2(\theta) + \cos^2(\theta) = 1 $.

So,

$ \left( \frac{1}{2} \right)^2 + \cos^2(\theta) = 1 $

$ \frac{1}{4} + \cos^2(\theta) = 1 $

$ \cos^2(\theta) = 1 – \frac{1}{4} $

$ \cos^2(\theta) = \frac{3}{4} $

$ \cos(\theta) = \pm \sqrt{\frac{3}{4}} $

Since $ \theta $ is in the first quadrant, $ \cos(\theta) $ is positive:

$ \cos(\theta) = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} $

Given that $ sin( heta) = frac{1}{2} $ and $ heta $ is in the first quadrant, we use the Pythagorean identity:

$ sin^2( heta) + cos^2( heta) = 1 $

Substitute $ sin( heta) = frac{1}{2} $:

$ left( frac{1}{2}
ight)^2 + cos^2( heta) = 1 $

$ frac{1}{4} + cos^2( heta) = 1 $

$ cos^2( heta) = 1 – frac{1}{4} = frac{3}{4} $

$ cos( heta) = frac{sqrt{3}}{2} $

Given $ sin( heta) = frac{1}{2} $, find $ cos( heta) $ using:

$ sin^2( heta) + cos^2( heta) = 1 $

$ cos( heta) = frac{sqrt{3}}{2} $

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