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Answer 1 To find the derivative of $ \cos(x^2) $ with respect to $ x $, we use the chain rule:\n$ \frac{d}{dx} \cos(u) = -\sin(u) \cdot \frac{du}{dx} $\nHere, let $ u = x^2 $. Then:\n$ \frac{du}{dx} = \frac{d}{dx}(x^2) = 2x $\nNow apply the chain...
Answer 1 To determine the tangent values for the primary angles on the unit circle, we need to evaluate the tangent function at txt1 txt1 txt1, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$ and $2\pi$.$...
Answer 1 To find the sine, cosine, and tangent of an angle using the unit circle, follow these steps:1. Locate the angle on the unit circle.2. Identify the coordinates $(x, y)$ of the point where the terminal side of the angle intersects the unit...
Answer 1 First, recall that the unit circle has a radius of 1. For the angle $ \theta = \frac{\pi}{3} $, we use the definitions of sine and cosine: $ x = \cos(\theta) $ $ y = \sin(\theta) $ When $ \theta = \frac{\pi}{3} $, we have: $ x =...
Answer 1 Given $\theta = \frac{5\pi}{4}$, we need to find the values of $\sin(\theta)$ and $\cos(\theta)$ on the unit circle.The angle $\frac{5\pi}{4}$ is in the third quadrant where both sine and cosine are negative.In the third quadrant, for an...
Answer 1 To compute the integral of $\cos^2(t)$ on the unit circle, we can use the double-angle identity for cosine:$\cos^2(t) = \frac{1 + \cos(2t)}{2}$Now, integrate:$\int_0^{2\pi} \cos^2(t) \, dt = \int_0^{2\pi} \frac{1 + \cos(2t)}{2} \,...