Trigonometry Examples

$\sin (\frac{30}{2}) = \frac{1}{m}$

Step 1

Rewrite the equation as $\frac{1}{m} = \sin (\frac{30}{2})$ .

$\frac{1}{m} = \sin (\frac{30}{2})$

Step 2

Simplify both sides.

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Step 2.1

Divide $30$ by $2$ .

$\frac{1}{m} = \sin (15)$

Step 2.2

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 2.2.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{1}{m} = \sin (45 - 30)$

Step 2.2.2

Separate negation.

$\frac{1}{m} = \sin (45 - (30))$

Step 2.2.3

Apply the difference of angles identity.

$\frac{1}{m} = \sin (45) \cos (30) - \cos (45) \sin (30)$

Step 2.2.4

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

$\frac{1}{m} = \frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)$

Step 2.2.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1}{m} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)$

Step 2.2.6

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

$\frac{1}{m} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)$

Step 2.2.7

The exact value of $\sin (30)$ 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 2.2.8

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

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Step 2.2.8.1

Simplify each term.

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Step 2.2.8.1.1

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

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Step 2.2.8.1.1.1

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 2.2.8.1.1.2

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 2.2.8.1.1.3

Multiply $2$ by $3$ .

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

Step 2.2.8.1.1.4

Multiply $2$ by $2$ .

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

Step 2.2.8.1.2

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

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Step 2.2.8.1.2.1

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

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

Step 2.2.8.1.2.2

Multiply $2$ by $2$ .

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

Step 2.2.8.2

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 $m$ .

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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.

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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.

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Step 4.3.2.1

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

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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.

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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 $4$ .

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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:

$m = 3.86370330 \dots$