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Answer 1 To solve for $x$ such that $\cos(x) = -\frac{1}{2}$ and $\sin(x)$ is negative on the unit circle, follow these steps: 1. Identify the angles where $\cos(x) = -\frac{1}{2}$. This occurs at $x = \frac{2\pi}{3}$ and $x = \frac{4\pi}{3}$ in...
Answer 1 To find the exact values of sine and cosine for the angle $\frac{5\pi}{4}$, we start by determining in which quadrant the angle lies. The angle $\frac{5\pi}{4}$ is in the third quadrant because $\frac{5\pi}{4} > \pi$ and $\frac{5\pi}{4} <...
Answer 1 To find $\cos(210^{\circ})$ and $\sin(210^{\circ})$, we start by converting the angle to radians:$210^{\circ} = 210 \cdot \frac{\pi}{180} = \frac{7\pi}{6}$The reference angle for $\frac{7\pi}{6}$ is $30^{\circ}$ or $\frac{\pi}{6}$.The...
Answer 1 $\text{The unit circle has the equation } x^2 + y^2 = 1.$$\text{To find the intersection with the x-axis, we set } y = 0.$$x^2 + 0^2 = 1$$x^2 = 1$$x = \pm 1.$$\text{Thus, the coordinates are } (1, 0) \text{ and } (-1, 0).$Answer 2 $ ext{The...
Answer 1 Given a point $P$ on the unit circle at an angle $\theta$, we can determine the coordinates of $P$ as follows: $ P(\cos(\theta), \sin(\theta)) $ The length of the line segment from $P$ to the origin is simply the radius of the unit circle,...
Answer 1 Let's consider the point (\frac{\sqrt{3}}{2}, \, \frac{1}{2}) on the unit circle. This point lies in the first quadrant and has coordinates (cos(\theta), sin(\theta)). We need to find the angle \theta that corresponds to this point. Using...