Explore the unit circle and its relationship to angles, radians, trigonometric ratios, and coordinates in the coordinate plane.
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Answer 1 To determine the $\cos$ and $\sin$ values for an angle of $45^\circ$ on the unit circle, follow these steps:1. Convert the angle from degrees to radians: $45^\circ = \frac{\pi}{4}$ radians.2. On the unit circle, the coordinates of a point...
Answer 1 First, we recognize that the unit circle is a circle with a radius of 1 centered at the origin (0, 0) in the Cartesian coordinate system. The angle \(-\frac{\pi}{2}\) radians corresponds to a rotation in the clockwise direction from the...
Answer 1 To find the exact values of the trigonometric functions for the angle $ \theta = \frac{5\pi}{4} $ using the unit circle, follow these steps:1. Locate the angle $ \theta = \frac{5\pi}{4} $ on the unit circle. This angle corresponds to $...
Answer 1 Consider the unit circle centered at the origin. Let point $P$ have coordinates $(x, y)$ on the unit circle such that the angle $\theta$ formed by the positive x-axis and the radius to $P$ is in the fourth quadrant. If the secant and tangent...
Answer 1 Let's consider the unit circle equation flipped along y=x: $x^2 + y^2 = 1$ becomes $y = x \cdot \sqrt{1 - x^2}$. Given a point where the x-coordinate is $\frac{1}{2}$, find the corresponding y-coordinate. Since the point lies on the flipped...
Answer 1 Let $\theta$ be an angle on the unit circle such that $\theta = \frac{7\pi}{6}$. Determine the exact values of $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$. The angle $\theta = \frac{7\pi}{6}$ is in the third quadrant. The reference...