Trigonometry Examples

tan (165)

$\tan (165)$

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 (15)

$- \tan (15)$

Step 2

Split

15

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

- tan (45 - 30)

$- \tan (45 - 30)$

Step 3

Separate negation.

- tan (45 - (30))

$- \tan (45 - (30))$

Step 4

Apply the difference of angles identity.

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

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

Step 5

The exact value of

tan (45)

$\tan (45)$ is

1

$1$ .

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

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

Step 6

The exact value of

tan (30)

$\tan (30)$ is

\frac{\sqrt{3}}{3}

$\frac{\sqrt{3}}{3}$ .

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

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

Step 7

The exact value of

tan (45)

$\tan (45)$ is

1

$1$ .

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

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

Step 8

The exact value of

tan (30)

$\tan (30)$ is

\frac{\sqrt{3}}{3}

$\frac{\sqrt{3}}{3}$ .

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

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

Step 9

Simplify

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

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

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Step 9.1

Multiply the numerator and denominator of the fraction by

3

$3$ .

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Step 9.1.1

Multiply

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

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

\frac{3}{3}

$\frac{3}{3}$ .

- ⎛ ⎜ ⎝ \frac{3}{3} \cdot \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}} ⎞ ⎟ ⎠

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

Step 9.1.2

Combine.

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

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

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

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

Step 9.2

Apply the distributive property.

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

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

Step 9.3

Cancel the common factor of

3

$3$ .

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Step 9.3.1

Move the leading negative in

- \frac{\sqrt{3}}{3}

$- \frac{\sqrt{3}}{3}$ into the numerator.

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

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

Step 9.3.2

Cancel the common factor.

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

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

Step 9.3.3

Rewrite the expression.

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

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

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

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

Step 9.4

Multiply

3

$3$ by

1

$1$ .

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

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

Step 9.5

Simplify the denominator.

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Step 9.5.1

Multiply

3

$3$ by

1

$1$ .

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

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

Step 9.5.2

Cancel the common factor of

3

$3$ .

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Step 9.5.2.1

Factor

3

$3$ out of

3 \cdot 1

$3 \cdot 1$ .

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

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

Step 9.5.2.2

Cancel the common factor.

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

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

Step 9.5.2.3

Rewrite the expression.

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

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

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

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

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

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

Step 9.6

Multiply

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

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

\frac{3 - \sqrt{3}}{3 - \sqrt{3}}

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

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

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

Step 9.7

Multiply

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

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

\frac{3 - \sqrt{3}}{3 - \sqrt{3}}

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

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

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

Step 9.8

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}}

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

Step 9.9

Simplify.

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

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

Step 9.10

Simplify the numerator.

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Step 9.10.1

Raise

3 - \sqrt{3}

$3 - \sqrt{3}$ to the power of

1

$1$ .

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

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

Step 9.10.2

Raise

3 - \sqrt{3}

$3 - \sqrt{3}$ to the power of

1

$1$ .

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

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

Step 9.10.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 9.10.4

Add

1

$1$ and

1

$1$ .

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

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

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

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

Step 9.11

Rewrite

{(3 - \sqrt{3})}^{2}

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

(3 - \sqrt{3}) (3 - \sqrt{3})

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

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

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

Step 9.12

Expand

(3 - \sqrt{3}) (3 - \sqrt{3})

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

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Step 9.12.1

Apply the distributive property.

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

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

Step 9.12.2

Apply the distributive property.

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

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

Step 9.12.3

Apply the distributive property.

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

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

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

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

Step 9.13

Simplify and combine like terms.

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Step 9.13.1

Simplify each term.

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Step 9.13.1.1

Multiply

3

$3$ by

3

$3$ .

- \frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}

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

Step 9.13.1.2

Multiply

- 1

$- 1$ by

3

$3$ .

- \frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}

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

Step 9.13.1.3

Multiply

3

$3$ by

- 1

$- 1$ .

- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}$

Step 9.13.1.4

Multiply

- \sqrt{3} (- \sqrt{3})

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

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Step 9.13.1.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}$

Step 9.13.1.4.2

Multiply

\sqrt{3}

$\sqrt{3}$ by

1

$1$ .

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

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

Step 9.13.1.4.3

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6}

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6}$

Step 9.13.1.4.4

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6}

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6}$

Step 9.13.1.4.5

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}$

Step 9.13.1.4.6

Add

1

$1$ and

1

$1$ .

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

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

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

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

Step 9.13.1.5

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 9.13.1.5.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6}

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

Step 9.13.1.5.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 9.13.1.5.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 9.13.1.5.4

Cancel the common factor of

$2$ .

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Step 9.13.1.5.4.1

Cancel the common factor.

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

Step 9.13.1.5.4.2

Rewrite the expression.

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6}$

Step 9.13.1.5.5

Evaluate the exponent.

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

Step 9.13.2

Add

$9$ and

$3$ .

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

Step 9.13.3

Subtract

$3 \sqrt{3}$ from

$- 3 \sqrt{3}$ .

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

Step 9.14

Cancel the common factor of

$12 - 6 \sqrt{3}$ and

$6$ .

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Step 9.14.1

Factor

$6$ out of

$12$ .

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

Step 9.14.2

Factor

$6$ out of

$- 6 \sqrt{3}$ .

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

Step 9.14.3

Factor

$6$ out of

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

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

Step 9.14.4

Cancel the common factors.

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Step 9.14.4.1

Factor

$6$ out of

$6$ .

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

Step 9.14.4.2

Cancel the common factor.

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

Step 9.14.4.3

Rewrite the expression.

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

Step 9.14.4.4

Divide

$2 - \sqrt{3}$ by

$1$ .

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

Step 9.15

Apply the distributive property.

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

Step 9.16

Multiply

$- 1$ by

$2$ .

$- 2 - - \sqrt{3}$

Step 9.17

Multiply

$- - \sqrt{3}$ .

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Step 9.17.1

Multiply

$- 1$ by

$- 1$ .

$- 2 + 1 \sqrt{3}$

Step 9.17.2

Multiply

$\sqrt{3}$ by

$1$ .

$- 2 + \sqrt{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- 2 + \sqrt{3}$

Decimal Form:

$- 0.26794919 \dots$