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 coordinates of points on the unit circle that satisfy the equation $ \cos^2(\theta) - \sin^2(\theta) = 0 $:First, we recall the Pythagorean identity: $ \cos^2(\theta) + \sin^2(\theta) = 1 $Given the equation: $...
Answer 1 $$ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$ Solution Steps: Determine the angle and the unit circle: The angle $( \frac{\pi}{4} )$ corresponds to 45° on the unit circle, where the coordinates are $(...
Answer 1 To find the coordinates of the point on the unit circle where the inverse sine function, $ \sin^{-1}(x) $, is equal to $ \frac{1}{2} $:We need to solve the equation:$ \sin^{-1}(y) = \frac{1}{2} $The angle whose sine is $ \frac{1}{2} $ is:$...
Answer 1 Given that $\cos(\theta) = -\frac{1}{\sqrt{2}}$ and $\theta$ is in the third quadrant, we start by using the identity:$\sec(\theta) = \frac{1}{\cos(\theta)}$Substituting the given value:$\sec(\theta) = \frac{1}{-\frac{1}{\sqrt{2}}}$We...
Answer 1 To find the sine and cosine values at $ t = \frac{\pi}{4} $ on the unit circle, we use the following values:$ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} $$ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} $Thus, the sine and...
Answer 1 To find the coordinates on the unit circle for $ \theta = \frac{\pi}{4} $, we use the unit circle properties.For $ \theta = \frac{\pi}{4} $, the coordinates are given by:$ (\cos(\frac{\pi}{4}), \sin(\frac{\pi}{4})) $From trigonometric...