Calculus Examples

$\cos (\sqrt{\frac{1}{2}})$

Step 1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\cos (\frac{\sqrt{1}}{\sqrt{2}})$

Step 2

Any root of $1$ is $1$ .

$\cos (\frac{1}{\sqrt{2}})$

Step 3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Evaluate the exponent.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.76024459 \dots$