Trigonometry Examples

$\tan (105)$

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 $\tan (105)$ is $- 2 - \sqrt{3}$ .

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Step 2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$- \tan (75) \cdot \frac{π}{180}$ radians

Step 2.2

Split $75$ into two angles where the values of the six trigonometric functions are known.

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

Step 2.3

Apply the sum of angles identity.

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

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

Step 2.8.1.2

Combine.

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

Step 2.8.2

Apply the distributive property.

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

Step 2.8.3

Cancel the common factor of $3$ .

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Step 2.8.3.1

Cancel the common factor.

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

Step 2.8.3.2

Rewrite the expression.

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

Step 2.8.4

Multiply $3$ by $1$ .

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

Step 2.8.5

Simplify the denominator.

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Step 2.8.5.1

Multiply $3$ by $1$ .

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

Step 2.8.5.2

Multiply $- 1$ by $1$ .

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

Step 2.8.5.3

Cancel the common factor of $3$ .

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Step 2.8.5.3.1

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

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

Step 2.8.5.3.2

Cancel the common factor.

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

Step 2.8.5.3.3

Rewrite the expression.

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

Step 2.8.6

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

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

Step 2.8.7

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

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

Step 2.8.8

Expand the denominator using the FOIL method.

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

Step 2.8.9

Simplify.

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

Step 2.8.10

Simplify the numerator.

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Step 2.8.10.1

Reorder terms.

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

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

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

Step 2.8.10.4

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 2.8.10.5

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

Move $3$ to the left of $\sqrt{3}$ .

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

Step 2.8.13.1.3

Combine using the product rule for radicals.

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

Step 2.8.13.1.4

Multiply $3$ by $3$ .

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

Step 2.8.13.1.5

Rewrite $9$ as $3^{2}$ .

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

Step 2.8.13.1.6

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

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

Add $3 \sqrt{3}$ and $3 \sqrt{3}$ .

$- \frac{12 + 6 \sqrt{3}}{6} \cdot \frac{π}{180}$ 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 2.8.15

Apply the distributive property.

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

Step 2.8.16

Multiply $- 1$ by $2$ .

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

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

Step 4.2

Factor $2$ out of $180$ .

$2 \cdot (- 1 \frac{π}{2 \cdot 90}) - \sqrt{3} \frac{π}{180}$ radians

Step 4.3

Cancel the common factor.

$2 \cdot (- 1 \frac{π}{2 \cdot 90}) - \sqrt{3} \frac{π}{180}$ radians

Step 4.4

Rewrite the expression.

$- 1 \frac{π}{90} - \sqrt{3} \frac{π}{180}$ radians

Step 5

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

$- 1 \frac{π}{90} - \frac{π \sqrt{3}}{180}$ radians

Step 6

Rewrite $- 1 \frac{π}{90}$ as $- \frac{π}{90}$ .

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