Answer 1 To find $\cot(\theta)$, we need to determine the appropriate angle $2\theta$. Given that the y-coordinate of $2\theta$ is $-\frac{\sqrt{3}}{2}$, we know that $2\theta$ corresponds to $240^\circ$ or $300^\circ$ in the unit circle. 1. For...

Answer 1 To find $ \tan \left( \frac{4\pi}{3} \right) $ on the unit circle, we note that $ \frac{4\pi}{3} $ radians is in the third quadrant.In the third quadrant, both sine and cosine are negative. The reference angle for $ \frac{4\pi}{3} $ is $...

Answer 1 Given the point on the unit circle represented by $e^{i\theta}$, we begin by expressing this in terms of its Cartesian coordinates:\[ e^{i\theta} = \cos(\theta) + i\sin(\theta) \]When the unit circle is flipped over the real axis, the...

Answer 1 The angle 45 degrees is equivalent to $\frac{\pi}{4}$ radians.On the unit circle, the coordinates for an angle of $45^\circ$ or $\frac{\pi}{4}$ radians are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.Thus, the cosine of $45^\circ$ is...

Answer 1 First, we need to find the reference angle for $\frac{4\pi}{3}$. The angle $\frac{4\pi}{3}$ is in the third quadrant.The reference angle is $\pi - (\frac{4\pi}{3} - \pi) = \frac{\pi}{3}$.In the third quadrant, the tangent function is...