Explore the unit circle and its relationship to angles, radians, trigonometric ratios, and coordinates in the coordinate plane.
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Answer 1 Given the point $ P = \left( -\frac{3}{5}, -\frac{4}{5} \right) $ on the unit circle, find the exact values of $ \sin(2\theta) $ and $ \cos(2\theta) $. The point $ P $ is in Quadrant III. To find $ \theta $, we use the definitions of sine...
Answer 1 To determine the sine, cosine, and tangent of $\theta = 30^\circ$ on the unit circle, we first need to recall the unit circle values for this angle. For $\theta = 30^\circ$: $\sin(30^\circ) = \frac{1}{2}$ $\cos(30^\circ) =...
Answer 1 Given $\sec(\theta) = 2$, we know $\sec(\theta) = \frac{1}{\cos(\theta)}$. So, $\frac{1}{\cos(\theta)} = 2$ implies $\cos(\theta) = \frac{1}{2}$. The cosine of $\theta$ is positive, so $\theta$ must be in the first or fourth quadrant....
Answer 1 To find the exact values of the trigonometric functions for the angle $\frac{7\pi}{6}$:1. Find the reference angle by subtracting $\pi$: $\frac{7\pi}{6} - \pi = \frac{7\pi}{6} - \frac{6\pi}{6} = \frac{\pi}{6}$2. Determine the coordinates on...
Answer 1 Given an angle of $\theta = \frac{5\pi}{6}$, determine the values of $\sin(\theta)$ and $\cos(\theta)$ using the unit circle.First, recognize that $\theta = \frac{5\pi}{6}$ is located in the second quadrant.In the second quadrant, the sine...
Answer 1 First, locate $ \theta = \frac{5\pi}{6} $ on the unit circle. This angle is in the second quadrant.In the second quadrant, the sine function is positive and the cosine function is negative. The reference angle for $ \theta = \frac{5\pi}{6} $...