Trigonometry Examples

$\arccos (x) + \arcsin (\frac{\sqrt{2}}{2}) = \frac{π}{2}$

Step 1

Simplify the left side.

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Step 1.1

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

$\arccos (x) + \frac{π}{4} = \frac{π}{2}$

Step 2

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

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Step 2.1

Subtract $\frac{π}{4}$ from both sides of the equation.

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

Step 2.2

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

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

Step 2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

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

Step 2.3.2

Multiply $2$ by $2$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Move $2$ to the left of $π$ .

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

Step 2.5.2

Subtract $π$ from $2 π$ .

$\arccos (x) = \frac{π}{4}$

Step 3

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

$x = \cos (\frac{π}{4})$

Step 4

Simplify the right side.

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Step 4.1

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.70710678 \dots$