Trigonometry Examples

$\sin (\frac{\frac{π}{6}}{2}) = \frac{1}{m}$

Step 1

Rewrite the equation as $\frac{1}{m} = \sin (\frac{\frac{π}{6}}{2})$ .

$\frac{1}{m} = \sin (\frac{\frac{π}{6}}{2})$

Step 2

Simplify both sides.

Tap for more steps...

Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{m} = \sin (\frac{π}{6} \cdot \frac{1}{2})$

Step 2.2

Multiply $\frac{π}{6} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 2.2.1

Multiply $\frac{π}{6}$ by $\frac{1}{2}$ .

$\frac{1}{m} = \sin (\frac{π}{6 \cdot 2})$

Step 2.2.2

Multiply $6$ by $2$ .

$\frac{1}{m} = \sin (\frac{π}{12})$

Step 2.3

The exact value of $\sin (\frac{π}{12})$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

Tap for more steps...

Step 2.3.1

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

$\frac{1}{m} = \sin (\frac{π}{4} - \frac{π}{6})$

Step 2.3.2

Apply the difference of angles identity.

$\frac{1}{m} = \sin (\frac{π}{4}) \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})$

Step 2.3.3

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

$\frac{1}{m} = \frac{\sqrt{2}}{2} \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})$

Step 2.3.4

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

$\frac{1}{m} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (\frac{π}{4}) \sin (\frac{π}{6})$

Step 2.3.5

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

$\frac{1}{m} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (\frac{π}{6})$

Step 2.3.6

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

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

Step 2.3.7

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

Tap for more steps...

Step 2.3.7.1

Simplify each term.

Tap for more steps...

Step 2.3.7.1.1

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

Tap for more steps...

Step 2.3.7.1.1.1

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

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

Step 2.3.7.1.1.2

Combine using the product rule for radicals.

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

Step 2.3.7.1.1.3

Multiply $2$ by $3$ .

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

Step 2.3.7.1.1.4

Multiply $2$ by $2$ .

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

Step 2.3.7.1.2

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

Tap for more steps...

Step 2.3.7.1.2.1

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

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

Step 2.3.7.1.2.2

Multiply $2$ by $2$ .

$\frac{1}{m} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$

Step 2.3.7.2

Combine the numerators over the common denominator.

$\frac{1}{m} = \frac{\sqrt{6} - \sqrt{2}}{4}$

Step 3

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$1 \cdot 4 = m (\sqrt{6} - \sqrt{2})$

Step 4

Solve the equation for $m$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $m (\sqrt{6} - \sqrt{2}) = 1 \cdot 4$ .

$m (\sqrt{6} - \sqrt{2}) = 1 \cdot 4$

Step 4.2

Multiply $4$ by $1$ .

$m (\sqrt{6} - \sqrt{2}) = 4$

Step 4.3

Divide each term in $m (\sqrt{6} - \sqrt{2}) = 4$ by $\sqrt{6} - \sqrt{2}$ and simplify.

Tap for more steps...

Step 4.3.1

Divide each term in $m (\sqrt{6} - \sqrt{2}) = 4$ by $\sqrt{6} - \sqrt{2}$ .

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

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

Cancel the common factor of $\sqrt{6} - \sqrt{2}$ .

Tap for more steps...

Step 4.3.2.1.1

Cancel the common factor.

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

Step 4.3.2.1.2

Divide $m$ by $1$ .

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

Step 4.3.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1

Multiply $\frac{4}{\sqrt{6} - \sqrt{2}}$ by $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$ .

$m = \frac{4}{\sqrt{6} - \sqrt{2}} \cdot \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$

Step 4.3.3.2

Multiply $\frac{4}{\sqrt{6} - \sqrt{2}}$ by $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$ .

$m = \frac{4 (\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2}) (\sqrt{6} + \sqrt{2})}$

Step 4.3.3.3

Expand the denominator using the FOIL method.

$m = \frac{4 (\sqrt{6} + \sqrt{2})}{{\sqrt{6}}^{2} + \sqrt{12} - \sqrt{12} - {\sqrt{2}}^{2}}$

Step 4.3.3.4

Simplify.

$m = \frac{4 (\sqrt{6} + \sqrt{2})}{4}$

Step 4.3.3.5

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.3.3.5.1

Cancel the common factor.

$m = \frac{4 (\sqrt{6} + \sqrt{2})}{4}$

Step 4.3.3.5.2

Divide $\sqrt{6} + \sqrt{2}$ by $1$ .

$m = \sqrt{6} + \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$m = \sqrt{6} + \sqrt{2}$

Decimal Form:

$m = 3.86370330 \dots$