/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Evaluate \(\cos \left(\frac{\pi}... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 25

Evaluate \(\cos \left(\frac{\pi}{6}+\cos ^{-1} \frac{3}{4}\right)\).

Short Answer

Expert verified
The short answer is that the expression \(\cos\left(\frac{\pi}{6}+\cos^{-1}\frac{3}{4}\right)\) evaluates to \(\frac{3\sqrt{3}-1}{8}\).

Step by step solution

01

Find \(\cos^{-1}\frac{3}{4}\)

First, find the value of \(\cos^{-1}\frac{3}{4}\). The angle whose cosine is \(\frac{3}{4}\) must lie in the first or fourth quadrant (since the cosine is positive), and it should be remembered that the principal value of an inverse trigonometric function always lies in the domains of the first and second quadrants. So, the angle is in the first quadrant. Using the Pythagorean theorem, for a right-angled triangle with adjacent side of length 3 and hypotenuse of length 4, the opposite side must have length 1. When we find the relevant angle in this triangle, we will know its cosine is \(\frac{3}{4}\). So, we can write: \[ \cos^{-1}\frac{3}{4} = \alpha \]
02

Add the angles

Now, add the angle \(\alpha\) to the angle \(\frac{\pi}{6}\). We have: \[ \frac{\pi}{6}+\alpha \]
03

Apply the Angle Sum Formula for Cosine

With the angle sum, we can apply the angle sum formula for cosine. This formula states: \[ \cos(A+B) = \cos A \cos B - \sin A \sin B \] In our case, \(A = \frac{\pi}{6}\) and \(B = \alpha\). Therefore, we'll have: \[ \cos\left(\frac{\pi}{6} + \alpha \right) = \cos\frac{\pi}{6} \cos\alpha - \sin\frac{\pi}{6} \sin\alpha \]
04

Substitute the known values

Now, we substitute the known values of cosine and find the values of sine for the given angles: \[ \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}, \ \sin\frac{\pi}{6} = \frac{1}{2}, \] and \[ \cos\alpha = \frac{3}{4}, \ \sin\alpha = \frac{1}{4} \] Now, we replace these values in our expression: \[ \cos\left(\frac{\pi}{6} + \alpha \right) = \left(\frac{\sqrt{3}}{2}\right)\left(\frac{3}{4}\right) - \left(\frac{1}{2}\right)\left(\frac{1}{4}\right) \]
05

Simplify the expression

Simplify the expression by performing the multiplications: \[ \cos\left(\frac{\pi}{6} + \alpha \right) = \frac{3\sqrt{3}}{8} - \frac{1}{8} \] Now, combine the terms to get the final answer: \[ \cos\left(\frac{\pi}{6} + \alpha \right) = \frac{3\sqrt{3} - 1}{8} \] Thus, \(\cos \left(\frac{\pi}{6}+\cos ^{-1} \frac{3}{4}\right) = \frac{3\sqrt{3}-1}{8}\).

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