Determine the value of $ cos( heta) $ when $ sin( heta) = frac{1}{2} $ in the unit circle

In the unit circle, when $ \sin(\theta) = \frac{1}{2} $, we need to determine $ \cos(\theta) $.

Since $ \sin(\theta) $ relates to the y-coordinate and $ \cos(\theta) $ relates to the x-coordinate in the unit circle, we use the Pythagorean identity:

$ \sin^2(\theta) + \cos^2(\theta) = 1 $

Given $ \sin(\theta) = \frac{1}{2} $, we can substitute:

$ \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} $

Taking the square root of both sides, we get:

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

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

To determine $ cos( heta) $ when $ sin( heta) = frac{1}{2} $, 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 $

Solve for $ cos^2( heta) $:

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

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

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

Using $ sin^2( heta) + cos^2( heta) = 1 $ and $ sin( heta) = frac{1}{2} $:

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

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