Trigonometry Examples

$\cos (\frac{5}{\sqrt{52}})$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 1.1.1

Factor $4$ out of $52$ .

$\cos (\frac{5}{\sqrt{4 (13)}})$

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

$\cos (\frac{5}{\sqrt{2^{2} \cdot 13}})$

Step 1.2

Pull terms out from under the radical.

$\cos (\frac{5}{2 \sqrt{13}})$

Step 2

Multiply $\frac{5}{2 \sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\cos (\frac{5}{2 \sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{5}{2 \sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\cos (\frac{5 \sqrt{13}}{2 \sqrt{13} \sqrt{13}})$

Step 3.2

Move $\sqrt{13}$ .

$\cos (\frac{5 \sqrt{13}}{2 (\sqrt{13} \sqrt{13})})$

Step 3.3

Raise $\sqrt{13}$ to the power of $1$ .

$\cos (\frac{5 \sqrt{13}}{2 ({\sqrt{13}}^{1} \sqrt{13})})$

Step 3.4

Raise $\sqrt{13}$ to the power of $1$ .

$\cos (\frac{5 \sqrt{13}}{2 ({\sqrt{13}}^{1} {\sqrt{13}}^{1})})$

Step 3.5

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

$\cos (\frac{5 \sqrt{13}}{2 {\sqrt{13}}^{1 + 1}})$

Step 3.6

Add $1$ and $1$ .

$\cos (\frac{5 \sqrt{13}}{2 {\sqrt{13}}^{2}})$

Step 3.7

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\cos (\frac{5 \sqrt{13}}{2 {(13^{\frac{1}{2}})}^{2}})$

Step 3.7.2

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

$\cos (\frac{5 \sqrt{13}}{2 \cdot 13^{\frac{1}{2} \cdot 2}})$

Step 3.7.3

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

$\cos (\frac{5 \sqrt{13}}{2 \cdot 13^{\frac{2}{2}}})$

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

$\cos (\frac{5 \sqrt{13}}{2 \cdot 13^{\frac{2}{2}}})$

Step 3.7.4.2

Rewrite the expression.

$\cos (\frac{5 \sqrt{13}}{2 \cdot 13^{1}})$

Step 3.7.5

Evaluate the exponent.

$\cos (\frac{5 \sqrt{13}}{2 \cdot 13})$

Step 4

Multiply $2$ by $13$ .

$\cos (\frac{5 \sqrt{13}}{26})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\cos (\frac{5 \sqrt{13}}{26})$

Decimal Form:

$0.76909315 \dots$