Trigonometry Examples

tan (165)

$\tan (165)$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case,

165

$165$ can be split into

120 + 45

$120 + 45$ .

tan (120 + 45)

$\tan (120 + 45)$

Step 2

Use the sum formula for tangent to simplify the expression. The formula states that

tan (A + B) = \frac{tan (A) + tan (B)}{1 - tan (A) tan (B)}

$\tan (A + B) = \frac{\tan (A) + \tan (B)}{1 - \tan (A) \tan (B)}$ .

\frac{tan (120) + tan (45)}{1 - tan (120) tan (45)}

$\frac{\tan (120) + \tan (45)}{1 - \tan (120) \tan (45)}$

Step 3

Simplify the numerator.

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Step 3.1

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.

\frac{- tan (60) + tan (45)}{1 - tan (120) tan (45)}

$\frac{- \tan (60) + \tan (45)}{1 - \tan (120) \tan (45)}$

Step 3.2

The exact value of

tan (60)

$\tan (60)$ is

\sqrt{3}

$\sqrt{3}$ .

\frac{- \sqrt{3} + tan (45)}{1 - tan (120) tan (45)}

$\frac{- \sqrt{3} + \tan (45)}{1 - \tan (120) \tan (45)}$

Step 3.3

The exact value of

tan (45)

$\tan (45)$ is

1

$1$ .

\frac{- \sqrt{3} + 1}{1 - tan (120) tan (45)}

$\frac{- \sqrt{3} + 1}{1 - \tan (120) \tan (45)}$

\frac{- \sqrt{3} + 1}{1 - tan (120) tan (45)}

$\frac{- \sqrt{3} + 1}{1 - \tan (120) \tan (45)}$

Step 4

Simplify the denominator.

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Step 4.1

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.

\frac{- \sqrt{3} + 1}{1 - - tan (60) tan (45)}

$\frac{- \sqrt{3} + 1}{1 - - \tan (60) \tan (45)}$

Step 4.2

The exact value of

tan (60)

$\tan (60)$ is

\sqrt{3}

$\sqrt{3}$ .

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

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

Step 4.3

Multiply

- - \sqrt{3}

$- - \sqrt{3}$ .

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Step 4.3.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 4.3.2

Multiply

\sqrt{3}

$\sqrt{3}$ by

1

$1$ .

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

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

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

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

Step 4.4

The exact value of

tan (45)

$\tan (45)$ is

1

$1$ .

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

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

Step 4.5

Multiply

\sqrt{3}

$\sqrt{3}$ by

1

$1$ .

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

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

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

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

Step 5

Multiply

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

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

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

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

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

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

Step 6

Combine fractions.

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Step 6.1

Multiply

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

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

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

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

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

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

Step 6.2

Expand the denominator using the FOIL method.

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

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

Step 6.3

Simplify.

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

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

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

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

Step 7

Simplify the numerator.

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Step 7.1

Reorder terms.

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

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

Step 7.2

Raise

1 - \sqrt{3}

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

1

$1$ .

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

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

Step 7.3

Raise

1 - \sqrt{3}

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

1

$1$ .

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

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

Step 7.4

Use the power rule

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

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

Step 7.5

Add

$1$ and

$1$ .

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

Step 8

Rewrite

${(1 - \sqrt{3})}^{2}$ as

$(1 - \sqrt{3}) (1 - \sqrt{3})$ .

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

Step 9

Expand

$(1 - \sqrt{3}) (1 - \sqrt{3})$ using the FOIL Method.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Apply the distributive property.

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

Step 10

Simplify and combine like terms.

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Step 10.1

Simplify each term.

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Step 10.1.1

Multiply

$1$ by

$1$ .

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

Step 10.1.2

Multiply

$- \sqrt{3}$ by

$1$ .

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

Step 10.1.3

Multiply

$- 1$ by

$1$ .

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

Step 10.1.4

Multiply

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

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Step 10.1.4.1

Multiply

$- 1$ by

$- 1$ .

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

Step 10.1.4.2

Multiply

$\sqrt{3}$ by

$1$ .

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

Step 10.1.4.3

Raise

$\sqrt{3}$ to the power of

$1$ .

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

Step 10.1.4.4

Raise

$\sqrt{3}$ to the power of

$1$ .

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

Step 10.1.4.5

Use the power rule

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

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

Step 10.1.4.6

Add

$1$ and

$1$ .

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

Step 10.1.5

Rewrite

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

$3$ .

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Step 10.1.5.1

Use

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

$\sqrt{3}$ as

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

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

Step 10.1.5.2

Apply the power rule and multiply exponents,

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

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

Step 10.1.5.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 10.1.5.4

Cancel the common factor of

$2$ .

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Step 10.1.5.4.1

Cancel the common factor.

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

Step 10.1.5.4.2

Rewrite the expression.

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

Step 10.1.5.5

Evaluate the exponent.

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

Step 10.2

Add

$1$ and

$3$ .

$\frac{4 - \sqrt{3} - \sqrt{3}}{- 2}$

Step 10.3

Subtract

$\sqrt{3}$ from

$- \sqrt{3}$ .

$\frac{4 - 2 \sqrt{3}}{- 2}$

Step 11

Cancel the common factor of

$4 - 2 \sqrt{3}$ and

$- 2$ .

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Step 11.1

Factor

$2$ out of

$4$ .

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

Step 11.2

Factor

$2$ out of

$- 2 \sqrt{3}$ .

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

Step 11.3

Factor

$2$ out of

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

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

Step 11.4

Move the negative one from the denominator of

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

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

Step 12

Rewrite

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

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

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

Step 13

Apply the distributive property.

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

Step 14

Multiply

$- 1$ by

$2$ .

$- 2 - - \sqrt{3}$

Step 15

Multiply

$- - \sqrt{3}$ .

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Step 15.1

Multiply

$- 1$ by

$- 1$ .

$- 2 + 1 \sqrt{3}$

Step 15.2

Multiply

$\sqrt{3}$ by

$1$ .

$- 2 + \sqrt{3}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$- 2 + \sqrt{3}$

Decimal Form:

$- 0.26794919 \dots$