Trigonometry Examples

cot (\frac{π}{12})

$\cot (\frac{π}{12})$

Step 1

Split

\frac{π}{12}

$\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\cot (\frac{π}{4} - \frac{π}{6})$

Step 2

Apply the difference of angles identity.

$\frac{\cot (\frac{π}{4}) \cot (\frac{π}{6}) + 1}{\cot (\frac{π}{6}) - \cot (\frac{π}{4})}$

Step 3

The exact value of

$\cot (\frac{π}{4})$ is

$1$ .

$\frac{1 \cot (\frac{π}{6}) + 1}{\cot (\frac{π}{6}) - \cot (\frac{π}{4})}$

Step 4

The exact value of

$\cot (\frac{π}{6})$ is

$\sqrt{3}$ .

$\frac{1 \sqrt{3} + 1}{\cot (\frac{π}{6}) - \cot (\frac{π}{4})}$

Step 5

The exact value of

$\cot (\frac{π}{6})$ is

$\sqrt{3}$ .

$\frac{1 \sqrt{3} + 1}{\sqrt{3} - \cot (\frac{π}{4})}$

Step 6

The exact value of

$\cot (\frac{π}{4})$ is

$1$ .

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

Step 7

Simplify

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

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Step 7.1

Multiply

$\sqrt{3}$ by

$1$ .

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

Step 7.2

Multiply

$- 1$ by

$1$ .

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

Step 7.3

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 7.4

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 7.5

Expand the denominator using the FOIL method.

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

Step 7.6

Simplify.

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

Step 7.7

Simplify the numerator.

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Step 7.7.1

Raise

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

$1$ .

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

Step 7.7.2

Raise

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

$1$ .

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

Step 7.7.3

Use the power rule

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

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

Step 7.7.4

Add

$1$ and

$1$ .

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

Step 7.8

Simplify

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

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Step 7.8.1

Rewrite

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

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

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

Step 7.8.2

Expand

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

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Step 7.8.2.1

Apply the distributive property.

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

Step 7.8.2.2

Apply the distributive property.

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

Step 7.8.2.3

Apply the distributive property.

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

Step 7.8.3

Simplify and combine like terms.

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Step 7.8.3.1

Simplify each term.

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Step 7.8.3.1.1

Combine using the product rule for radicals.

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

Step 7.8.3.1.2

Multiply

$3$ by

$3$ .

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

Step 7.8.3.1.3

Rewrite

$9$ as

$3^{2}$ .

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

Step 7.8.3.1.4

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 7.8.3.1.5

Multiply

$\sqrt{3}$ by

$1$ .

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

Step 7.8.3.1.6

Multiply

$\sqrt{3}$ by

$1$ .

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

Step 7.8.3.1.7

Multiply

$1$ by

$1$ .

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

Step 7.8.3.2

Add

$3$ and

$1$ .

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

Step 7.8.3.3

Add

$\sqrt{3}$ and

$\sqrt{3}$ .

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

Step 7.9

Cancel the common factor of

$4 + 2 \sqrt{3}$ and

$2$ .

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Step 7.9.1

Factor

$2$ out of

$4$ .

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

Step 7.9.2

Factor

$2$ out of

$2 \sqrt{3}$ .

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

Step 7.9.3

Factor

$2$ out of

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

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

Step 7.9.4

Cancel the common factors.

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Step 7.9.4.1

Factor

$2$ out of

$2$ .

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

Step 7.9.4.2

Cancel the common factor.

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

Step 7.9.4.3

Rewrite the expression.

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

Step 7.9.4.4

Divide

$2 + \sqrt{3}$ by

$1$ .

$2 + \sqrt{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$2 + \sqrt{3}$

Decimal Form:

$3.73205080 \dots$