Trigonometry Examples

$\arccos (x - \frac{1}{\sqrt{2}}) = \frac{π}{4}$

Step 1

Take the inverse arccosine of both sides of the equation to extract $x$ from inside the arccosine.

$x - \frac{1}{\sqrt{2}} = \cos (\frac{π}{4})$

Step 2

Simplify the left side.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

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

$x - (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) = \cos (\frac{π}{4})$

Step 2.1.2

Combine and simplify the denominator.

Tap for more steps...

Step 2.1.2.1

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

$x - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} = \cos (\frac{π}{4})$

Step 2.1.2.2

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

$x - \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} = \cos (\frac{π}{4})$

Step 2.1.2.3

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

$x - \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} = \cos (\frac{π}{4})$

Step 2.1.2.4

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

$x - \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} = \cos (\frac{π}{4})$

Step 2.1.2.5

Add $1$ and $1$ .

$x - \frac{\sqrt{2}}{{\sqrt{2}}^{2}} = \cos (\frac{π}{4})$

Step 2.1.2.6

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

Tap for more steps...

Step 2.1.2.6.1

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

$x - \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} = \cos (\frac{π}{4})$

Step 2.1.2.6.2

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

$x - \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} = \cos (\frac{π}{4})$

Step 2.1.2.6.3

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

$x - \frac{\sqrt{2}}{2^{\frac{2}{2}}} = \cos (\frac{π}{4})$

Step 2.1.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.2.6.4.1

Cancel the common factor.

$x - \frac{\sqrt{2}}{2^{\frac{2}{2}}} = \cos (\frac{π}{4})$

Step 2.1.2.6.4.2

Rewrite the expression.

$x - \frac{\sqrt{2}}{2^{1}} = \cos (\frac{π}{4})$

Step 2.1.2.6.5

Evaluate the exponent.

$x - \frac{\sqrt{2}}{2} = \cos (\frac{π}{4})$

Step 3

Simplify the right side.

Tap for more steps...

Step 3.1

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

$x - \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}$

Step 4

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 4.1

Add $\frac{\sqrt{2}}{2}$ to both sides of the equation.

$x = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}$

Step 4.2

Combine the numerators over the common denominator.

$x = \frac{\sqrt{2} + \sqrt{2}}{2}$

Step 4.3

Add $\sqrt{2}$ and $\sqrt{2}$ .

$x = \frac{2 \sqrt{2}}{2}$

Step 4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.1

Cancel the common factor.

$x = \frac{2 \sqrt{2}}{2}$

Step 4.4.2

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

$x = \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{2}$

Decimal Form:

$x = 1.41421356 \dots$