Trigonometry Examples

$\tan (15)$

Step 1

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

Step 2

The exact value of $\tan (15)$ is $2 - \sqrt{3}$ .

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Step 2.1

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

$\tan (45 - 30) \cdot \frac{π}{180}$ radians

Step 2.2

Separate negation.

$\tan (45 - (30)) \cdot \frac{π}{180}$ radians

Step 2.3

Apply the difference of angles identity.

$\frac{\tan (45) - \tan (30)}{1 + \tan (45) \tan (30)} \cdot \frac{π}{180}$ radians

Step 2.4

The exact value of $\tan (45)$ is $1$ .

$\frac{1 - \tan (30)}{1 + \tan (45) \tan (30)} \cdot \frac{π}{180}$ radians

Step 2.5

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

$\frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (45) \tan (30)} \cdot \frac{π}{180}$ radians

Step 2.6

The exact value of $\tan (45)$ is $1$ .

$\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (30)} \cdot \frac{π}{180}$ radians

Step 2.7

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

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

Step 2.8

Simplify $\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}$ .

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Step 2.8.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 2.8.1.1

Multiply $\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}$ by $\frac{3}{3}$ .

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

Step 2.8.1.2

Combine.

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

Step 2.8.2

Apply the distributive property.

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

Step 2.8.3

Cancel the common factor of $3$ .

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Step 2.8.3.1

Move the leading negative in $- \frac{\sqrt{3}}{3}$ into the numerator.

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

Step 2.8.3.2

Cancel the common factor.

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

Step 2.8.3.3

Rewrite the expression.

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

Step 2.8.4

Multiply $3$ by $1$ .

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

Step 2.8.5

Simplify the denominator.

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Step 2.8.5.1

Multiply $3$ by $1$ .

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

Step 2.8.5.2

Cancel the common factor of $3$ .

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Step 2.8.5.2.1

Factor $3$ out of $3 \cdot 1$ .

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

Step 2.8.5.2.2

Cancel the common factor.

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

Step 2.8.5.2.3

Rewrite the expression.

$\frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{π}{180}$ radians

Step 2.8.6

Multiply $\frac{3 - \sqrt{3}}{3 + \sqrt{3}}$ by $\frac{3 - \sqrt{3}}{3 - \sqrt{3}}$ .

$\frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}} \cdot \frac{π}{180}$ radians

Step 2.8.7

Multiply $\frac{3 - \sqrt{3}}{3 + \sqrt{3}}$ by $\frac{3 - \sqrt{3}}{3 - \sqrt{3}}$ .

$\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{(3 + \sqrt{3}) (3 - \sqrt{3})} \cdot \frac{π}{180}$ radians

Step 2.8.8

Expand the denominator using the FOIL method.

$\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}} \cdot \frac{π}{180}$ radians

Step 2.8.9

Simplify.

$\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6} \cdot \frac{π}{180}$ radians

Step 2.8.10

Simplify the numerator.

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Step 2.8.10.1

Raise $3 - \sqrt{3}$ to the power of $1$ .

$\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6} \cdot \frac{π}{180}$ radians

Step 2.8.10.2

Raise $3 - \sqrt{3}$ to the power of $1$ .

$\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6} \cdot \frac{π}{180}$ radians

Step 2.8.10.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(3 - \sqrt{3})}^{1 + 1}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.10.4

Add $1$ and $1$ .

$\frac{{(3 - \sqrt{3})}^{2}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.11

Rewrite ${(3 - \sqrt{3})}^{2}$ as $(3 - \sqrt{3}) (3 - \sqrt{3})$ .

$\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6} \cdot \frac{π}{180}$ radians

Step 2.8.12

Expand $(3 - \sqrt{3}) (3 - \sqrt{3})$ using the FOIL Method.

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Step 2.8.12.1

Apply the distributive property.

$\frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6} \cdot \frac{π}{180}$ radians

Step 2.8.12.2

Apply the distributive property.

$\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6} \cdot \frac{π}{180}$ radians

Step 2.8.12.3

Apply the distributive property.

$\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13

Simplify and combine like terms.

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Step 2.8.13.1

Simplify each term.

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Step 2.8.13.1.1

Multiply $3$ by $3$ .

$\frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.2

Multiply $- 1$ by $3$ .

$\frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.3

Multiply $3$ by $- 1$ .

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 2.8.13.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.4.4

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.4.6

Add $1$ and $1$ .

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.8.13.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.5.4

Cancel the common factor of $2$ .

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Step 2.8.13.1.5.4.1

Cancel the common factor.

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.5.4.2

Rewrite the expression.

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.1.5.5

Evaluate the exponent.

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.2

Add $9$ and $3$ .

$\frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.13.3

Subtract $3 \sqrt{3}$ from $- 3 \sqrt{3}$ .

$\frac{12 - 6 \sqrt{3}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.14

Cancel the common factor of $12 - 6 \sqrt{3}$ and $6$ .

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Step 2.8.14.1

Factor $6$ out of $12$ .

$\frac{6 \cdot 2 - 6 \sqrt{3}}{6} \cdot \frac{π}{180}$ radians

Step 2.8.14.2

Factor $6$ out of $- 6 \sqrt{3}$ .

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

Step 2.8.14.3

Factor $6$ out of $6 (2) + 6 (- \sqrt{3})$ .

$\frac{6 (2 - \sqrt{3})}{6} \cdot \frac{π}{180}$ radians

Step 2.8.14.4

Cancel the common factors.

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Step 2.8.14.4.1

Factor $6$ out of $6$ .

$\frac{6 (2 - \sqrt{3})}{6 (1)} \cdot \frac{π}{180}$ radians

Step 2.8.14.4.2

Cancel the common factor.

$\frac{6 (2 - \sqrt{3})}{6 \cdot 1} \cdot \frac{π}{180}$ radians

Step 2.8.14.4.3

Rewrite the expression.

$\frac{2 - \sqrt{3}}{1} \cdot \frac{π}{180}$ radians

Step 2.8.14.4.4

Divide $2 - \sqrt{3}$ by $1$ .

$(2 - \sqrt{3}) \cdot \frac{π}{180}$ radians

Step 3

Apply the distributive property.

$2 (\frac{π}{180}) - \sqrt{3} \frac{π}{180}$ radians

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Factor $2$ out of $180$ .

$2 (\frac{π}{2 (90)}) - \sqrt{3} \frac{π}{180}$ radians

Step 4.2

Cancel the common factor.

$2 (\frac{π}{2 \cdot 90}) - \sqrt{3} \frac{π}{180}$ radians

Step 4.3

Rewrite the expression.

$\frac{π}{90} - \sqrt{3} \frac{π}{180}$ radians

Step 5

Combine $\frac{π}{180}$ and $\sqrt{3}$ .

$\frac{π}{90} - \frac{π \sqrt{3}}{180}$ radians