Trigonometry Examples

$\sin (\frac{π}{4} + x)$

Step 1

Use the sum formula for sine to simplify the expression. The formula states that $\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$ .

$\sin (\frac{π}{4}) \cos (x) + \cos (\frac{π}{4}) \sin (x)$

Step 2

Remove parentheses.

$\sin (\frac{π}{4}) \cos (x) + \cos (\frac{π}{4}) \sin (x)$

Step 3

Simplify each term.

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Step 3.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} \cos (x) + \cos (\frac{π}{4}) \sin (x)$

Step 3.2

Combine $\frac{\sqrt{2}}{2}$ and $\cos (x)$ .

$\frac{\sqrt{2} \cos (x)}{2} + \cos (\frac{π}{4}) \sin (x)$

Step 3.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2} \cos (x)}{2} + \frac{\sqrt{2}}{2} \sin (x)$

Step 3.4

Combine $\frac{\sqrt{2}}{2}$ and $\sin (x)$ .

$\frac{\sqrt{2} \cos (x)}{2} + \frac{\sqrt{2} \sin (x)}{2}$

Step 4

Simplify.

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Step 4.1

Combine the numerators over the common denominator.

$\frac{\sqrt{2} \cos (x) + \sqrt{2} \sin (x)}{2}$

Step 4.2

Factor $\sqrt{2}$ out of $\sqrt{2} \cos (x) + \sqrt{2} \sin (x)$ .

$\frac{\sqrt{2} (\cos (x) + \sin (x))}{2}$