Determine the cosine of the angle $t$ on the unit circle when the sine of $t$ is $frac{1}{2}$

To find the cosine of the angle $t$ on the unit circle when the sine of $t$ is $\frac{1}{2}$, we can use the Pythagorean identity:

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

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Given that $\sin(t) = \frac{1}{2}$, we substitute and solve for $\cos(t)$:

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

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

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$ \cos^2(t) = 1 – \frac{1}{4} $

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

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Therefore, $\cos(t)$ can be:

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

To find the cosine of the angle $t$ when $sin(t) = frac{1}{2}$, use the Pythagorean identity:

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

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

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

Simplify and solve for $cos(t)$:

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

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

Thus, $cos(t)$ is:

$ cos(t) = pmfrac{sqrt{3}}{2} $

To find $cos(t)$ when $sin(t) = frac{1}{2}$:

Use the identity:

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

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

$ cos(t) = pmfrac{sqrt{3}}{2} $

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