Trigonometry Examples

$\frac{\sqrt{1 - \cos (\frac{30}{2})}}{2}$

Step 1

Cancel the common factor of $30$ and $2$ .

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Step 1.1

Factor $2$ out of $30$ .

$\frac{\sqrt{1 - \cos (\frac{2 \cdot 15}{2})}}{2}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $2$ .

$\frac{\sqrt{1 - \cos (\frac{2 \cdot 15}{2 (1)})}}{2}$

Step 1.2.2

Cancel the common factor.

$\frac{\sqrt{1 - \cos (\frac{2 \cdot 15}{2 \cdot 1})}}{2}$

Step 1.2.3

Rewrite the expression.

$\frac{\sqrt{1 - \cos (\frac{15}{1})}}{2}$

Step 1.2.4

Divide $15$ by $1$ .

$\frac{\sqrt{1 - \cos (15)}}{2}$

Step 2

Simplify the numerator.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Separate negation.

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

Step 2.1.3

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.1.6

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

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

Step 2.1.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

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

Step 2.1.8

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

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Step 2.1.8.1

Simplify each term.

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Step 2.1.8.1.1

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

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Step 2.1.8.1.1.1

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

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

Step 2.1.8.1.1.2

Combine using the product rule for radicals.

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

Step 2.1.8.1.1.3

Multiply $2$ by $3$ .

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

Step 2.1.8.1.1.4

Multiply $2$ by $2$ .

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

Step 2.1.8.1.2

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

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Step 2.1.8.1.2.1

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

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

Step 2.1.8.1.2.2

Multiply $2$ by $2$ .

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

Step 2.1.8.2

Combine the numerators over the common denominator.

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

Step 2.2

Write $1$ as a fraction with a common denominator.

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

Rewrite $\sqrt{\frac{4 - \sqrt{6} - \sqrt{2}}{4}}$ as $\frac{\sqrt{4 - \sqrt{6} - \sqrt{2}}}{\sqrt{4}}$ .

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

Step 2.6

Simplify the denominator.

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Step 2.6.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.6.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

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

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Step 4.1

Multiply $\frac{\sqrt{4 - \sqrt{6} - \sqrt{2}}}{2}$ by $\frac{1}{2}$ .

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

Step 4.2

Multiply $2$ by $2$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.09229595 \dots$