$ ext{Find the cosine values of the angles on the unit circle}$

Given the angle $\theta = \frac{5\pi}{3}$, we need to find the cosine value.

The unit circle coordinates at an angle $\theta$ are given by $(\cos(\theta), \sin(\theta))$. For $\theta = \frac{5\pi}{3}$, the angle is in the fourth quadrant where the cosine is positive and sine is negative.

Using reference angles, we can see that $\frac{5\pi}{3}$ is equivalent to $-\frac{\pi}{3}$ or $2\pi – \frac{\pi}{3}$. Thus, the cosine value is:

$\cos\left(\frac{5\pi}{3}\right) = \cos\left(2\pi – \frac{\pi}{3}\right) = \cos\left(-\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right)$

From the unit circle, we know that $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$. Therefore,

$\cos\left(\frac{5\pi}{3}\right) = \frac{1}{2}$

Given the angle $ heta = frac{7pi}{4}$, we need to find the cosine value.

The unit circle coordinates at an angle $ heta$ are given by $(cos( heta), sin( heta))$. For $ heta = frac{7pi}{4}$, the angle is in the fourth quadrant where the cosine is positive and sine is negative.

Using reference angles, we can see that $frac{7pi}{4}$ is equivalent to $-frac{pi}{4}$ or $2pi – frac{pi}{4}$. Thus, the cosine value is:

$cosleft(frac{7pi}{4}
ight) = cosleft(2pi – frac{pi}{4}
ight) = cosleft(-frac{pi}{4}
ight) = cosleft(frac{pi}{4}
ight)$

From the unit circle, we know that $cosleft(frac{pi}{4}
ight) = frac{sqrt{2}}{2}$. Therefore,

$cosleft(frac{7pi}{4}
ight) = frac{sqrt{2}}{2}$

Given the angle $ heta = frac{11pi}{6}$, we need to find the cosine value.

The unit circle coordinates at an angle $ heta$ are given by $(cos( heta), sin( heta))$. For $ heta = frac{11pi}{6}$, we use reference angles.

Using reference angles, we can see that $frac{11pi}{6}$ is equivalent to $2pi – frac{pi}{6}$. Thus, the cosine value is:

$cosleft(frac{11pi}{6}
ight) = cosleft(2pi – frac{pi}{6}
ight) = cosleft(frac{pi}{6}
ight)$

From the unit circle, we know that $cosleft(frac{pi}{6}
ight) = frac{sqrt{3}}{2}$. Therefore,

$cosleft(frac{11pi}{6}
ight) = frac{sqrt{3}}{2}$

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