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Chapter 6: Problem 32

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

Short Answer

Expert verified
The formula for \(\sin\left(\theta - \frac{\pi}{4}\right)\) is \(\frac{\sqrt{2}}{2}\left(\sin\theta - \cos\theta\right)\).

Step by step solution

01

Apply Sum and Difference Formula for Sine

To find the formula for \(\sin\left(\theta - \frac{\pi}{4}\right)\), we need to apply the sine sum and difference formula: \(\sin(a - b) = \sin a \cos b - \cos a \sin b\). Here, \(a = \theta\) and \(b = \frac{\pi}{4}\). So the formula becomes: \[ \sin\left(\theta - \frac{\pi}{4}\right) = \sin\theta\cos\frac{\pi}{4} - \cos\theta\sin\frac{\pi}{4} \]
02

Find the values for \(\sin\frac{\pi}{4}\) and \(\cos\frac{\pi}{4}\)

We know that \(\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}\) and \(\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}\).
03

Substitute the values in original equation

Now that we have found the values of \(\sin\frac{\pi}{4}\) and \(\cos\frac{\pi}{4}\), we can substitute them in the formula derived in Step 1: \[ \sin\left(\theta - \frac{\pi}{4}\right) = \sin\theta\cos\frac{\pi}{2^4} - \cos\theta\sin\frac{\pi}{4}=\sin\theta\left(\frac{\sqrt{2}}{2}\right) - \cos\theta\left(\frac{\sqrt{2}}{2}\right) \]
04

Simplify the expression

Simplifying the above expression, we get: \[ \sin\left(\theta - \frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\left(\sin\theta - \cos\theta\right) \] So the formula for \(\sin\left(\theta - \frac{\pi}{4}\right)\) is \(\frac{\sqrt{2}}{2}\left(\sin\theta - \cos\theta\right)\).

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