Trigonometry Examples

$\sec (\frac{π}{4}) + 2 \csc (\frac{π}{3})$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

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

Step 1.2

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

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

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

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

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

Step 1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 1.3.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 1.3.4

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

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

Step 1.3.5

Add $1$ and $1$ .

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

Step 1.3.6

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

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Step 1.3.6.1

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

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

Step 1.3.6.2

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

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

Step 1.3.6.3

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

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

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.

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

Step 1.3.6.4.2

Rewrite the expression.

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

Step 1.3.6.5

Evaluate the exponent.

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

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

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

$\sqrt{2} + 2 \csc (\frac{π}{3})$

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

Combine and simplify the denominator.

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Step 1.7.1

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

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

Step 1.7.2

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

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

Step 1.7.3

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

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

Step 1.7.4

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

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

Step 1.7.5

Add $1$ and $1$ .

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

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

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

Step 1.7.6.2

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

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

Step 1.7.6.3

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

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

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.

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

Step 1.7.6.4.2

Rewrite the expression.

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

Step 1.7.6.5

Evaluate the exponent.

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

Step 1.8

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

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Step 1.8.1

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

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

Step 1.8.2

Multiply $2$ by $2$ .

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

Step 2

To write $\sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3

Combine fractions.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Move $3$ to the left of $\sqrt{2}$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$3.72361463 \dots$