Determine the values of $cos( heta)$ and $sin( heta)$ using the unit circle when Define the unit circle in trigonometry ≤ heta ≤ 2pi$ and $ heta$ is a solution to the equation $ an( heta) = sqrt{3}$

The equation $\tan(\theta) = \sqrt{3}$ implies that:

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \sqrt{3}$

This happens at $\theta = \frac{\pi}{3}$ and $\theta = \frac{4\pi}{3}$ within the interval

txt1

txt1

txt1

≤ \theta ≤ 2\pi$.

At $\theta = \frac{\pi}{3}$:

$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}, \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$

At $\theta = \frac{4\pi}{3}$:

$\cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}, \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}$

Thus, the values are:

$\theta = \frac{\pi}{3}: \cos(\theta) = \frac{1}{2}, \sin(\theta) = \frac{\sqrt{3}}{2}$

$\theta = \frac{4\pi}{3}: \cos(\theta) = -\frac{1}{2}, \sin(\theta) = -\frac{\sqrt{3}}{2}$

Given $ an( heta) = sqrt{3}$, we identify that:

$ an( heta) = frac{sin( heta)}{cos( heta)} = sqrt{3}$

Within the interval

txt2

txt2

txt2

≤ heta ≤ 2pi$, this is true for:

$ heta = frac{pi}{3}$

$cosleft(frac{pi}{3}
ight) = frac{1}{2}, sinleft(frac{pi}{3}
ight) = frac{sqrt{3}}{2}$

$ heta = frac{4pi}{3}$

$cosleft(frac{4pi}{3}
ight) = -frac{1}{2}, sinleft(frac{4pi}{3}
ight) = -frac{sqrt{3}}{2}$

For $ an( heta) = sqrt{3}$, $ heta$ can be:

$ heta = frac{pi}{3}$

$cosleft(frac{pi}{3}
ight) = frac{1}{2}, sinleft(frac{pi}{3}
ight) = frac{sqrt{3}}{2}$

Or

$ heta = frac{4pi}{3}$

$cosleft(frac{4pi}{3}
ight) = -frac{1}{2}, sinleft(frac{4pi}{3}
ight) = -frac{sqrt{3}}{2}$

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