Evaluate arcsin(sin(pi/12))

$\arcsin (\sin (\frac{π}{12}))$

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\arcsin (\sin (\frac{π}{4} - \frac{π}{6}))$

Apply the difference of angles identity.

$\arcsin (\sin (\frac{π}{4}) \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6}))$

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

$\arcsin (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6}))$

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\arcsin (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (\frac{π}{4}) \sin (\frac{π}{6}))$

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

$\arcsin (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (\frac{π}{6}))$

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

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

Simplify each term.

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

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

$\arcsin (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Multiply $2$ by $2$ .

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

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

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

$\arcsin (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})$

Multiply $2$ by $2$ .

$\arcsin (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$

Combine the numerators over the common denominator.

$\arcsin (\frac{\sqrt{6} - \sqrt{2}}{4})$

The result can be shown in multiple forms.

Exact Form:

$\arcsin (\frac{\sqrt{6} - \sqrt{2}}{4})$

Decimal Form:

$0.26179938 \dots$