Trigonometry Examples

$\cos (\frac{π}{12})$

Step 1

To convert radians to degrees, multiply by $\frac{180}{π}$ , since a full circle is $360 °$ or $2 π$ radians.

$(\cos (\frac{π}{12})) \cdot \frac{180 °}{π}$

Step 2

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

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Step 2.1

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

$\cos (\frac{π}{4} - \frac{π}{6}) \cdot \frac{180}{π}$

Step 2.2

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

$(\cos (\frac{π}{4}) \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6})) \cdot \frac{180}{π}$

Step 2.3

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

$(\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6})) \cdot \frac{180}{π}$

Step 2.4

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

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (\frac{π}{4}) \sin (\frac{π}{6})) \cdot \frac{180}{π}$

Step 2.5

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

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (\frac{π}{6})) \cdot \frac{180}{π}$

Step 2.6

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

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \frac{180}{π}$

Step 2.7

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

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

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

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Step 2.7.1.1.1

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

$(\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \frac{180}{π}$

Step 2.7.1.1.2

Combine using the product rule for radicals.

$(\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \frac{180}{π}$

Step 2.7.1.1.3

Multiply $2$ by $3$ .

$(\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \frac{180}{π}$

Step 2.7.1.1.4

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \frac{180}{π}$

Step 2.7.1.2

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

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Step 2.7.1.2.1

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

$(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) \cdot \frac{180}{π}$

Step 2.7.1.2.2

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) \cdot \frac{180}{π}$

Step 2.7.2

Combine the numerators over the common denominator.

$\frac{\sqrt{6} + \sqrt{2}}{4} \cdot \frac{180}{π}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $4$ .

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Step 3.1.1

Factor $4$ out of $180$ .

$\frac{\sqrt{6} + \sqrt{2}}{4} \cdot \frac{4 (45)}{π}$

Step 3.1.2

Cancel the common factor.

$\frac{\sqrt{6} + \sqrt{2}}{4} \cdot \frac{4 \cdot 45}{π}$

Step 3.1.3

Rewrite the expression.

$(\sqrt{6} + \sqrt{2}) \cdot \frac{45}{π}$

Step 3.2

Apply the distributive property.

$\sqrt{6} \frac{45}{π} + \sqrt{2} \frac{45}{π}$

Step 3.3

Combine $\sqrt{6}$ and $\frac{45}{π}$ .

$\frac{\sqrt{6} \cdot 45}{π} + \sqrt{2} \frac{45}{π}$

Step 3.4

Combine $\sqrt{2}$ and $\frac{45}{π}$ .

$\frac{\sqrt{6} \cdot 45}{π} + \frac{\sqrt{2} \cdot 45}{π}$

Step 4

Simplify each term.

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Step 4.1

Move $45$ to the left of $\sqrt{6}$ .

$\frac{45 \sqrt{6}}{π} + \frac{\sqrt{2} \cdot 45}{π}$

Step 4.2

Move $45$ to the left of $\sqrt{2}$ .

$\frac{45 \sqrt{6}}{π} + \frac{45 \sqrt{2}}{π}$

Step 5

$π$ is approximately equal to $3.14159265$ .

$\frac{45 \sqrt{6}}{3.14159265} + \frac{45 \sqrt{2}}{3.14159265}$

Step 6

Convert to a decimal.

$55.34347316 °$