Trigonometry Examples

$\cot (165)$

Step 1

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

$- \cot (15)$

Step 2

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

$- \cot (45 - 30)$

Step 3

Separate negation.

$- \cot (45 - (30))$

Step 4

Apply the difference of angles identity.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Multiply $- 1$ by $1$ .

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

Expand the denominator using the FOIL method.

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

Step 9.6

Simplify.

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

Step 9.7

Simplify the numerator.

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Step 9.7.1

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

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

Step 9.7.2

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

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

Step 9.7.3

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

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

Step 9.7.4

Add $1$ and $1$ .

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

Step 9.8

Simplify ${(\sqrt{3} + 1)}^{2}$ .

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Step 9.8.1

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

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

Step 9.8.2

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

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Step 9.8.2.1

Apply the distributive property.

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

Step 9.8.2.2

Apply the distributive property.

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

Step 9.8.2.3

Apply the distributive property.

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

Step 9.8.3

Simplify and combine like terms.

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Step 9.8.3.1

Simplify each term.

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Step 9.8.3.1.1

Combine using the product rule for radicals.

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

Step 9.8.3.1.2

Multiply $3$ by $3$ .

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

Step 9.8.3.1.3

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

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

Step 9.8.3.1.4

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

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

Step 9.8.3.1.5

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

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

Step 9.8.3.1.6

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

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

Step 9.8.3.1.7

Multiply $1$ by $1$ .

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

Step 9.8.3.2

Add $3$ and $1$ .

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

Step 9.8.3.3

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

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

Step 9.9

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

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Step 9.9.1

Factor $2$ out of $4$ .

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

Step 9.9.2

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

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

Step 9.9.3

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

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

Step 9.9.4

Cancel the common factors.

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Step 9.9.4.1

Factor $2$ out of $2$ .

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

Step 9.9.4.2

Cancel the common factor.

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

Step 9.9.4.3

Rewrite the expression.

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

Step 9.9.4.4

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

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

Step 9.10

Apply the distributive property.

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

Step 9.11

Multiply $- 1$ by $2$ .

$- 2 - \sqrt{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- 2 - \sqrt{3}$

Decimal Form:

$- 3.73205080 \dots$