Trigonometry Examples

$\cot (165)$

Step 1

Replace $\cot (165)$ with an equivalent expression $\frac{1}{\tan (165)}$ using the fundamental identities.

$\frac{1}{\tan (165)}$

Step 2

Use a sum or difference formula on the denominator.

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Step 2.1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $165$ can be split into $120 + 45$ .

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

Step 2.2

Use the sum formula for tangent to simplify the expression. The formula states that $\tan (A + B) = \frac{\tan (A) + \tan (B)}{1 - \tan (A) \tan (B)}$ .

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

Step 2.3

Simplify the numerator.

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Step 2.3.1

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.

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

Step 2.3.2

The exact value of $\tan (60)$ is $\sqrt{3}$ .

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

Step 2.3.3

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

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

Step 2.4

Simplify the denominator.

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Step 2.4.1

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.

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

Step 2.4.2

The exact value of $\tan (60)$ is $\sqrt{3}$ .

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

Step 2.4.3

Multiply $- - \sqrt{3}$ .

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Step 2.4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.4.3.2

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

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

Step 2.4.4

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

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

Step 2.4.5

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

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

Step 2.5

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

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

Step 2.6

Combine fractions.

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Step 2.6.1

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

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

Step 2.6.2

Expand the denominator using the FOIL method.

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

Step 2.6.3

Simplify.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Reorder terms.

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

Step 2.7.2

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

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

Step 2.7.3

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

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

Step 2.7.4

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

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

Step 2.7.5

Add $1$ and $1$ .

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

Step 2.8

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

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

Step 2.9

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

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Step 2.9.1

Apply the distributive property.

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

Step 2.9.2

Apply the distributive property.

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

Step 2.9.3

Apply the distributive property.

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

Step 2.10

Simplify and combine like terms.

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Step 2.10.1

Simplify each term.

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Step 2.10.1.1

Multiply $1$ by $1$ .

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

Step 2.10.1.2

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

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

Step 2.10.1.3

Multiply $- 1$ by $1$ .

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

Step 2.10.1.4

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

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Step 2.10.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.10.1.4.2

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

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

Step 2.10.1.4.3

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

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

Step 2.10.1.4.4

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

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

Step 2.10.1.4.5

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

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

Step 2.10.1.4.6

Add $1$ and $1$ .

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

Step 2.10.1.5

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

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Step 2.10.1.5.1

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

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

Step 2.10.1.5.2

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

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

Step 2.10.1.5.3

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

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

Step 2.10.1.5.4

Cancel the common factor of $2$ .

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Step 2.10.1.5.4.1

Cancel the common factor.

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

Step 2.10.1.5.4.2

Rewrite the expression.

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

Step 2.10.1.5.5

Evaluate the exponent.

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

Step 2.10.2

Add $1$ and $3$ .

$\frac{1}{\frac{4 - \sqrt{3} - \sqrt{3}}{- 2}}$

Step 2.10.3

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

$\frac{1}{\frac{4 - 2 \sqrt{3}}{- 2}}$

Step 2.11

Cancel the common factor of $4 - 2 \sqrt{3}$ and $- 2$ .

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Step 2.11.1

Factor $2$ out of $4$ .

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

Step 2.11.2

Factor $2$ out of $- 2 \sqrt{3}$ .

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

Step 2.11.3

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

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

Step 2.11.4

Move the negative one from the denominator of $\frac{2 - \sqrt{3}}{- 1}$ .

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

Step 2.12

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

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

Step 2.13

Apply the distributive property.

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

Step 2.14

Multiply $- 1$ by $2$ .

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

Step 2.15

Multiply $- - \sqrt{3}$ .

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Step 2.15.1

Multiply $- 1$ by $- 1$ .

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

Step 2.15.2

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

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

Step 3

Simplify.

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Step 3.1

Multiply $\frac{1}{- 2 + \sqrt{3}}$ by $\frac{- 2 - \sqrt{3}}{- 2 - \sqrt{3}}$ .

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

Step 3.2

Multiply $\frac{1}{- 2 + \sqrt{3}}$ by $\frac{- 2 - \sqrt{3}}{- 2 - \sqrt{3}}$ .

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

Step 3.3

Expand the denominator using the FOIL method.

$\frac{- 2 - \sqrt{3}}{4 + 2 \sqrt{3} + \sqrt{3} \cdot - 2 - {\sqrt{3}}^{2}}$

Step 3.4

Simplify.

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

Step 3.5

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

$- 2 - \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- 2 - \sqrt{3}$

Decimal Form:

$- 3.73205080 \dots$