/*! 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 31 Find a formula for \(\cos \left(... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 31

Find a formula for \(\cos \left(\theta+\frac{\pi}{4}\right)\).

Short Answer

Expert verified
The formula for \(\cos\left(\theta + \frac{\pi}{4}\right)\) is: \[\cos\left(\theta + \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\cos(\theta) - \frac{1}{\sqrt{2}}\sin(\theta).\]

Step by step solution

01

Recognize the angle addition formula

The angle addition formula for cosine is: \[\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B).\] In this problem, we will use this formula with \(A = \theta\) and \(B = \frac{\pi}{4}\).
02

Find the cosine and sine of \(\frac{\pi}{4}\)

We need to find the cosine and sine of the angle \(\frac{\pi}{4}\), which is equivalent to 45 degrees: \[\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\text{ and }\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}.\]
03

Apply the angle addition formula

Now, we can plug in the values we found in the previous step into the angle addition formula, as well as substitute \(\theta\) for \(A\) and \(\frac{\pi}{4}\) for \(B\). \[\cos\left(\theta + \frac{\pi}{4}\right) = \cos(\theta)\cos\left(\frac{\pi}{4}\right) - \sin(\theta)\sin\left(\frac{\pi}{4}\right).\]
04

Plug in the values and simplify

From step 2, we know \(\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\) and \(\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\). Substitute these values into the equation: \[\cos\left(\theta + \frac{\pi}{4}\right) = \cos(\theta)\left(\frac{1}{\sqrt{2}}\right) - \sin(\theta)\left(\frac{1}{\sqrt{2}}\right).\] Now we can simplify the expression by distributing the \(\frac{1}{\sqrt{2}}\) factor, getting: \[\cos\left(\theta + \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\cos(\theta) - \frac{1}{\sqrt{2}}\sin(\theta).\]
05

Final Answer

The formula for \(\cos\left(\theta + \frac{\pi}{4}\right)\) is: \[\cos\left(\theta + \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\cos(\theta) - \frac{1}{\sqrt{2}}\sin(\theta).\]

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