Trigonometry Examples

$\arccos (x) + 2 \arcsin (\frac{\sqrt{3}}{2}) = π$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

The exact value of $\arcsin (\frac{\sqrt{3}}{2})$ is $\frac{π}{3}$ .

$\arccos (x) + 2 \frac{π}{3} = π$

Step 1.1.2

Combine $2$ and $\frac{π}{3}$ .

$\arccos (x) + \frac{2 π}{3} = π$

Step 2

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

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Step 2.1

Subtract $\frac{2 π}{3}$ from both sides of the equation.

$\arccos (x) = π - \frac{2 π}{3}$

Step 2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\arccos (x) = π \cdot \frac{3}{3} - \frac{2 π}{3}$

Step 2.3

Combine $π$ and $\frac{3}{3}$ .

$\arccos (x) = \frac{π \cdot 3}{3} - \frac{2 π}{3}$

Step 2.4

Combine the numerators over the common denominator.

$\arccos (x) = \frac{π \cdot 3 - 2 π}{3}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Move $3$ to the left of $π$ .

$\arccos (x) = \frac{3 \cdot π - 2 π}{3}$

Step 2.5.2

Subtract $2 π$ from $3 π$ .

$\arccos (x) = \frac{π}{3}$

Step 3

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

$x = \cos (\frac{π}{3})$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$x = \frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}$

Decimal Form:

$x = 0.5$