Trigonometry Examples

$\sec (\frac{π}{6}) + 2 \cot (\frac{π}{3})$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

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

Step 1.2

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Add $1$ and $1$ .

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

Step 1.3.6

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

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Step 1.3.6.1

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

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

Step 1.3.6.2

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

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

Step 1.3.6.3

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

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

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

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

Step 1.3.6.4.2

Rewrite the expression.

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

Step 1.3.6.5

Evaluate the exponent.

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

Step 1.4

The exact value of $\cot (\frac{π}{3})$ is $\frac{1}{\sqrt{3}}$ .

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

Step 1.5

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

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

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

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

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

Step 1.6.2

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

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

Step 1.6.3

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

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

Step 1.6.4

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

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

Step 1.6.5

Add $1$ and $1$ .

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

Step 1.6.6

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

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Step 1.6.6.1

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

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

Step 1.6.6.2

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

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

Step 1.6.6.3

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

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

Step 1.6.6.4

Cancel the common factor of $2$ .

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Step 1.6.6.4.1

Cancel the common factor.

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

Step 1.6.6.4.2

Rewrite the expression.

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

Step 1.6.6.5

Evaluate the exponent.

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

Step 1.7

Combine $2$ and $\frac{\sqrt{3}}{3}$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$2.30940107 \dots$