Answer 1 To understand the unit circle, consider the following:1. Identify the key points on the unit circle where the angle is 0, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, and $\frac{\pi}{2}$. 2. Recall the coordinates of these points: $...

Answer 1 To find the values of $\sin$, $\cos$, and $\tan$ for an angle of 120 degrees, first convert the angle to radians: $120^\circ = \frac{120 \pi}{180} = \frac{2 \pi}{3}$ Next, locate the angle on the unit circle. The angle $\frac{2 \pi}{3}$ is...

Answer 1 To find the exact $\tan$ values at specific angles on the unit circle, consider the following:1. $\theta = \frac{\pi}{4}$ At this angle, $\tan(\theta) = \tan\left(\frac{\pi}{4}\right) = 1$2. $\theta = \frac{2\pi}{3}$ At this angle,...

Answer 1 Consider a complex number $z$ on the unit circle in the complex plane. The unit circle can be represented as $|z| = 1$. If $z = e^{i\theta}$, find the real part of $z$.Solution:Since $z = e^{i\theta}$, we can use Euler's formula, which...

Answer 1 Given the initial coordinates $(x, y)$ and angle $\theta$, the coordinates after rotating by $45^\circ$ counterclockwise can be found using the rotation matrix: $ \begin{bmatrix} \cos(45^\circ) & -\sin(45^\circ) \\ \sin(45^\circ) &...