Answer 1 To find the tangent of $45^\circ$ on the unit circle, we use the fact that $\tan \theta = \frac{\sin \theta}{\cos \theta}$. At $45^\circ$, $\sin 45^\circ = \frac{\sqrt{2}}{2}$ and $\cos 45^\circ = \frac{\sqrt{2}}{2}$. Therefore, $ \tan...
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Given a point on the unit circle where the secant of the angle is 3, find the angle in radians and degrees, and determine the corresponding coordinates on the unit circle
Answer 1 Given that $\sec \theta = 3$, we know that: $\sec \theta = \frac{1}{\cos \theta}$Solving for $\cos \theta$, we get:$\cos \theta = \frac{1}{3}$Using $\cos^{-1}(\frac{1}{3})$, we find:$\theta = \cos^{-1}(\frac{1}{3})$Converting to...
Find the sine and cosine of an angle given in radians on the unit circle
Answer 1 Given an angle \( \theta = \frac{\pi}{4} \), find the sine and cosine of the angle on the unit circle. Using the unit circle, the coordinates of the point at \( \theta = \frac{\pi}{4} \) are given by: \( (\cos(\frac{\pi}{4}),...
Find the values of sin(θ), cos(θ), and tan(θ) using the unit circle for θ = 135°
Answer 1 We start by locating the angle $\theta = 135°$ on the unit circle.Since $135°$ is in the second quadrant, we use the reference angle $45°$ to find the values. The coordinates of the point on the unit circle at this angle are $\left(...
Given the unit circle, find the length of the chord subtending an angle of θ at the center Explain the steps and provide the solution
Answer 1 To find the length of the chord subtending an angle $\theta$ at the center of the unit circle, we can use the formula for the chord length: $L = 2r \sin\left(\frac{\theta}{2}\right)$ Since the radius $r$ of the unit circle is 1, the formula...
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