Evaluate arccos(1/( square root of 2))

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 3

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

$\frac{π}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{4}$

Decimal Form:

$0.78539816 \dots$