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 find the angle that corresponds to the point $ \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) $ on the unit circle, we look at the coordinates.The x-coordinate is $ \frac{1}{2} $ and the y-coordinate is $ -\frac{\sqrt{3}}{2} $. These...
Answer 1 To find the values of $ \theta $ for which $ \tan(\theta) = 1 $ on the unit circle, we need to identify the angles where the tangent function is equal to 1.The tangent function is defined as the ratio of the sine and cosine functions:$...
Answer 1 To find the exact values of the trigonometric functions for the angle $ \frac{5\pi}{3} $ in the unit circle, we first note that:$ \frac{5\pi}{3} = 2\pi - \frac{\pi}{3} $This means the angle is located in the fourth quadrant.We can use the...
Answer 1 On the unit circle, angles are measured starting from the positive x-axis and moving counterclockwise.To find the location of $-\pi/2$:- Starting from the positive x-axis, move clockwise because the angle is negative.- $\pi/2$ is 90 degrees,...
Answer 1 Given the points $ (\cos(\theta), \sin(\theta))$, the origin $(0, 0)$, and $(1, 0)$, we need to find $ \theta $ such that they form a right-angled triangle.The distance between $(\cos(\theta), \sin(\theta))$ and $(1, 0)$ is:$ d =...
Answer 1 To calculate the area of a sector in a unit circle with a given central angle $ \theta $ (in radians), use the following formula: $ A = \frac{1}{2} \cdot r^2 \cdot \theta $ Since the radius $ r $ of a unit circle is 1, the formula simplifies...