Question Write the expression as a function of $x$ : $\cos\left(\frac{\pi}{4}-x\right)$ .

Studdy Solution

STEP 1

1. The expression $\cos \left(\frac{\pi}{4}-x\right)$ involves the cosine function, which is a trigonometric function.
2. The angle subtraction formula for cosine can be used to simplify the expression: $\cos(A-B) = \cos A \cos B + \sin A \sin B$ .
3. The values of $\cos \left(\frac{\pi}{4}\right)$ and $\sin \left(\frac{\pi}{4}\right)$ are known and equal to $\frac{\sqrt{2}}{2}$ .

STEP 2

1. Apply the cosine angle subtraction formula.
2. Substitute the known values of $\cos \left(\frac{\pi}{4}\right)$ and $\sin \left(\frac{\pi}{4}\right)$ .
3. Simplify the expression to get the function of $x$ .

STEP 3

Apply the cosine angle subtraction formula to $\cos \left(\frac{\pi}{4}-x\right)$ .
$\cos \left(\frac{\pi}{4}-x\right) = \cos \left(\frac{\pi}{4}\right) \cos(x) + \sin \left(\frac{\pi}{4}\right) \sin(x)$

STEP 4

Substitute the known values of $\cos \left(\frac{\pi}{4}\right)$ and $\sin \left(\frac{\pi}{4}\right)$ into the formula.
$\cos \left(\frac{\pi}{4}-x\right) = \frac{\sqrt{2}}{2} \cos(x) + \frac{\sqrt{2}}{2} \sin(x)$

STEP 5

Simplify the expression by combining the terms with a common factor.
$\cos \left(\frac{\pi}{4}-x\right) = \frac{\sqrt{2}}{2} (\cos(x) + \sin(x))$
The final expression as a function of $x$ , with no angle measure involved, is:
$\cos \left(\frac{\pi}{4}-x\right) = \frac{\sqrt{2}}{2} (\cos(x) + \sin(x))$

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