$Find the value of sec( heta) when the terminal point of angle heta lies on the unit circle at coordinates left( frac{1}{2}, frac{sqrt{3}}{2}
ight).$

Given the coordinates $\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)$ on the unit circle, we know that the x-coordinate represents $\cos(\theta)$. Therefore:

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

The secant function is the reciprocal of the cosine function:

$ \sec(\theta) = \frac{1}{\cos(\theta)} $

Substitute $ \cos(\theta)$ with $\frac{1}{2}$:

$ \sec(\theta) = \frac{1}{\frac{1}{2}} = 2 $

Therefore, the value of $ \sec(\theta)$ is 2.

First, identify the x-coordinate given on the unit circle:

$x = frac{1}{2}$

This x-coordinate represents the cosine of the angle $ heta$:

$ cos( heta) = frac{1}{2} $

We need to find $ sec( heta)$, which is defined as:

$ sec( heta) = frac{1}{cos( heta)} $

Substituting in the value of $cos( heta):$

$ sec( heta) = frac{1}{frac{1}{2}} = 2 $

Thus, the value of $ sec( heta)$ is 2.

Given the coordinates $left( frac{1}{2}, frac{sqrt{3}}{2}
ight)$:

$ cos( heta) = frac{1}{2} $

Then:

$ sec( heta) = frac{1}{cos( heta)} = frac{1}{frac{1}{2}} = 2 $

Hence, $ sec( heta)$ is 2.

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