Trigonometry Examples

$8 \sec (\frac{π}{6}) + 5 \cot (\frac{π}{3})$

Step 1

Simplify each term.

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Step 1.1

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

$8 \frac{2}{\sqrt{3}} + 5 \cot (\frac{π}{3})$

Step 1.2

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

$8 (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) + 5 \cot (\frac{π}{3})$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

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

$8 \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} + 5 \cot (\frac{π}{3})$

Step 1.3.2

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

$8 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} + 5 \cot (\frac{π}{3})$

Step 1.3.3

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

$8 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} + 5 \cot (\frac{π}{3})$

Step 1.3.4

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

$8 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} + 5 \cot (\frac{π}{3})$

Step 1.3.5

Add $1$ and $1$ .

$8 \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} + 5 \cot (\frac{π}{3})$

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

$8 \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} + 5 \cot (\frac{π}{3})$

Step 1.3.6.2

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

$8 \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} + 5 \cot (\frac{π}{3})$

Step 1.3.6.3

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

$8 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} + 5 \cot (\frac{π}{3})$

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.

$8 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} + 5 \cot (\frac{π}{3})$

Step 1.3.6.4.2

Rewrite the expression.

$8 \frac{2 \sqrt{3}}{3^{1}} + 5 \cot (\frac{π}{3})$

Step 1.3.6.5

Evaluate the exponent.

$8 \frac{2 \sqrt{3}}{3} + 5 \cot (\frac{π}{3})$

Step 1.4

Multiply $8 \frac{2 \sqrt{3}}{3}$ .

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Step 1.4.1

Combine $8$ and $\frac{2 \sqrt{3}}{3}$ .

$\frac{8 (2 \sqrt{3})}{3} + 5 \cot (\frac{π}{3})$

Step 1.4.2

Multiply $2$ by $8$ .

$\frac{16 \sqrt{3}}{3} + 5 \cot (\frac{π}{3})$

Step 1.5

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

$\frac{16 \sqrt{3}}{3} + 5 \frac{1}{\sqrt{3}}$

Step 1.6

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

$\frac{16 \sqrt{3}}{3} + 5 (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 1.7

Combine and simplify the denominator.

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Step 1.7.1

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

$\frac{16 \sqrt{3}}{3} + 5 \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 1.7.2

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

$\frac{16 \sqrt{3}}{3} + 5 \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 1.7.3

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

$\frac{16 \sqrt{3}}{3} + 5 \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 1.7.4

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

$\frac{16 \sqrt{3}}{3} + 5 \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 1.7.5

Add $1$ and $1$ .

$\frac{16 \sqrt{3}}{3} + 5 \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 1.7.6

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

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Step 1.7.6.1

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

$\frac{16 \sqrt{3}}{3} + 5 \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 1.7.6.2

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

$\frac{16 \sqrt{3}}{3} + 5 \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 1.7.6.3

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

$\frac{16 \sqrt{3}}{3} + 5 \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.7.6.4

Cancel the common factor of $2$ .

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Step 1.7.6.4.1

Cancel the common factor.

$\frac{16 \sqrt{3}}{3} + 5 \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.7.6.4.2

Rewrite the expression.

$\frac{16 \sqrt{3}}{3} + 5 \frac{\sqrt{3}}{3^{1}}$

Step 1.7.6.5

Evaluate the exponent.

$\frac{16 \sqrt{3}}{3} + 5 \frac{\sqrt{3}}{3}$

Step 1.8

Combine $5$ and $\frac{\sqrt{3}}{3}$ .

$\frac{16 \sqrt{3}}{3} + \frac{5 \sqrt{3}}{3}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{16 \sqrt{3} + 5 \sqrt{3}}{3}$

Step 2.2

Add $16 \sqrt{3}$ and $5 \sqrt{3}$ .

$\frac{21 \sqrt{3}}{3}$

Step 2.3

Cancel the common factor of $21$ and $3$ .

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Step 2.3.1

Factor $3$ out of $21 \sqrt{3}$ .

$\frac{3 (7 \sqrt{3})}{3}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 (7 \sqrt{3})}{3 (1)}$

Step 2.3.2.2

Cancel the common factor.

$\frac{3 (7 \sqrt{3})}{3 \cdot 1}$

Step 2.3.2.3

Rewrite the expression.

$\frac{7 \sqrt{3}}{1}$

Step 2.3.2.4

Divide $7 \sqrt{3}$ by $1$ .

$7 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$7 \sqrt{3}$

Decimal Form:

$12.12435565 \dots$