Expand Using Sum/Difference Formulas sin(15 degrees )

$\sin (15 °)$

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $15 °$ can be split into $60 - 45$ .

$\sin (60 - 45)$

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

$\sin (60) \cos (45) - \cos (60) \sin (45)$

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$(\frac{\sqrt{3}}{2}) \cos (45) - \cos (60) \sin (45)$

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$(\frac{\sqrt{3}}{2}) (\frac{\sqrt{2}}{2}) - \cos (60) \sin (45)$

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$(\frac{\sqrt{3}}{2}) (\frac{\sqrt{2}}{2}) - \frac{1}{2} \sin (45)$

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$(\frac{\sqrt{3}}{2}) (\frac{\sqrt{2}}{2}) - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$

Simplify each term.

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Multiply $(\frac{\sqrt{3}}{2}) (\frac{\sqrt{2}}{2})$ .

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Multiply $\frac{\sqrt{3}}{2}$ and $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$

Multiply $3$ by $2$ .

$\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$

Multiply $- \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Multiply $\frac{\sqrt{2}}{2}$ and $\frac{1}{2}$ .

$\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - \sqrt{2}}{4}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6} - \sqrt{2}}{4}$

Decimal Form:

$0.25881904 \dots$

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