Determine the cosine value of $-pi/3$ using the unit circle

First, recall that the unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. In the unit circle, the angle $\theta = -\pi/3$ is measured in the clockwise direction.

To find the cosine of $-\pi/3$, we can use the symmetry of the unit circle. The angle $-\pi/3$ is the same as $5\pi/3$ in the standard position (i.e., measured counterclockwise from the positive x-axis).

Cosine corresponds to the x-coordinate of the point on the unit circle. Thus, we need to find the x-coordinate of the point corresponding to $5\pi/3$.

At $5\pi/3$, the point on the unit circle is $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$. Therefore, the cosine of $-\pi/3$ is:

$\cos(-\pi/3) = \frac{1}{2}$

To determine the cosine value of $-pi/3$ using the unit circle, recall that angles measured in the clockwise direction are negative.

We can convert $-pi/3$ to a positive angle by adding $2pi$:

$-pi/3 + 2pi = frac{-pi + 6pi}{3} = frac{5pi}{3}$

Next, locate the angle $5pi/3$ on the unit circle. This angle lies in the fourth quadrant, where the cosine is positive.

The reference angle for $5pi/3$ is $pi/3$, which has a corresponding point on the unit circle of $left(frac{1}{2}, -frac{sqrt{3}}{2}
ight)$.

Therefore, the cosine of $-pi/3$ is:

$cos(-pi/3) = frac{1}{2}$

Considering the unit circle and the angle $-pi/3$:

$-pi/3 + 2pi = frac{5pi}{3}$

The point $left(frac{1}{2}, -frac{sqrt{3}}{2}
ight)$ corresponds to $5pi/3$.

Thus,

$cos(-pi/3) = frac{1}{2}$

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