Trigonometry Examples

$\tan (105 °)$

Step 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.

$- \tan (75)$

Step 2

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

$- \tan (30 + 45)$

Step 3

Apply the sum of angles identity.

$- \frac{\tan (30) + \tan (45)}{1 - \tan (30) \tan (45)}$

Step 4

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

$- \frac{\frac{\sqrt{3}}{3} + \tan (45)}{1 - \tan (30) \tan (45)}$

Step 5

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

$- \frac{\frac{\sqrt{3}}{3} + 1}{1 - \tan (30) \tan (45)}$

Step 6

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

$- \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \tan (45)}$

Step 7

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

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

Step 8

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

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Step 8.1

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

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Step 8.1.1

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

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

Step 8.1.2

Combine.

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Cancel the common factor of $3$ .

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Step 8.3.1

Cancel the common factor.

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

Step 8.3.2

Rewrite the expression.

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

Step 8.4

Multiply $3$ by $1$ .

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

Step 8.5

Simplify the denominator.

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Step 8.5.1

Multiply $3$ by $1$ .

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

Step 8.5.2

Multiply $- 1$ by $1$ .

$- \frac{\sqrt{3} + 3}{3 + 3 (- \frac{\sqrt{3}}{3})}$

Step 8.5.3

Cancel the common factor of $3$ .

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Step 8.5.3.1

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

$- \frac{\sqrt{3} + 3}{3 + 3 \frac{- \sqrt{3}}{3}}$

Step 8.5.3.2

Cancel the common factor.

$- \frac{\sqrt{3} + 3}{3 + 3 \frac{- \sqrt{3}}{3}}$

Step 8.5.3.3

Rewrite the expression.

$- \frac{\sqrt{3} + 3}{3 - \sqrt{3}}$

Step 8.6

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

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

Step 8.7

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

$- \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{(3 - \sqrt{3}) (3 + \sqrt{3})}$

Step 8.8

Expand the denominator using the FOIL method.

$- \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{9 + 3 \sqrt{3} - 3 \sqrt{3} - {\sqrt{3}}^{2}}$

Step 8.9

Simplify.

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

Step 8.10

Simplify the numerator.

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Step 8.10.1

Reorder terms.

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

Step 8.10.2

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

$- \frac{{(3 + \sqrt{3})}^{1} (3 + \sqrt{3})}{6}$

Step 8.10.3

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

$- \frac{{(3 + \sqrt{3})}^{1} {(3 + \sqrt{3})}^{1}}{6}$

Step 8.10.4

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

$- \frac{{(3 + \sqrt{3})}^{1 + 1}}{6}$

Step 8.10.5

Add $1$ and $1$ .

$- \frac{{(3 + \sqrt{3})}^{2}}{6}$

Step 8.11

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

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

Step 8.12

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

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Step 8.12.1

Apply the distributive property.

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

Step 8.12.2

Apply the distributive property.

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

Step 8.12.3

Apply the distributive property.

$- \frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6}$

Step 8.13

Simplify and combine like terms.

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Step 8.13.1

Simplify each term.

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Step 8.13.1.1

Multiply $3$ by $3$ .

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

Step 8.13.1.2

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

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

Step 8.13.1.3

Combine using the product rule for radicals.

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

Step 8.13.1.4

Multiply $3$ by $3$ .

$- \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{9}}{6}$

Step 8.13.1.5

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

$- \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3^{2}}}{6}$

Step 8.13.1.6

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

$- \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + 3}{6}$

Step 8.13.2

Add $9$ and $3$ .

$- \frac{12 + 3 \sqrt{3} + 3 \sqrt{3}}{6}$

Step 8.13.3

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

$- \frac{12 + 6 \sqrt{3}}{6}$

Step 8.14

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

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Step 8.14.1

Factor $6$ out of $12$ .

$- \frac{6 \cdot 2 + 6 \sqrt{3}}{6}$

Step 8.14.2

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

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

Step 8.14.3

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

$- \frac{6 (2 + \sqrt{3})}{6}$

Step 8.14.4

Cancel the common factors.

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Step 8.14.4.1

Factor $6$ out of $6$ .

$- \frac{6 (2 + \sqrt{3})}{6 (1)}$

Step 8.14.4.2

Cancel the common factor.

$- \frac{6 (2 + \sqrt{3})}{6 \cdot 1}$

Step 8.14.4.3

Rewrite the expression.

$- \frac{2 + \sqrt{3}}{1}$

Step 8.14.4.4

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

$- (2 + \sqrt{3})$

Step 8.15

Apply the distributive property.

$- 1 \cdot 2 - \sqrt{3}$

Step 8.16

Multiply $- 1$ by $2$ .

$- 2 - \sqrt{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- 2 - \sqrt{3}$

Decimal Form:

$- 3.73205080 \dots$