Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.3

To write $- \frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ .

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

Step 2.4.2

Multiply $2$ by $3$ .

$\arccos (x) = \frac{π \cdot 3}{6} - \frac{π}{3} \cdot \frac{2}{2}$

Step 2.4.3

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$\arccos (x) = \frac{π \cdot 3}{6} - \frac{π \cdot 2}{3 \cdot 2}$

Step 2.4.4

Multiply $3$ by $2$ .

$\arccos (x) = \frac{π \cdot 3}{6} - \frac{π \cdot 2}{6}$

Step 2.5

Combine the numerators over the common denominator.

$\arccos (x) = \frac{π \cdot 3 - π \cdot 2}{6}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Move $3$ to the left of $π$ .

$\arccos (x) = \frac{3 \cdot π - π \cdot 2}{6}$

Step 2.6.2

Multiply $2$ by $- 1$ .

$\arccos (x) = \frac{3 π - 2 π}{6}$

Step 2.6.3

Subtract $2 π$ from $3 π$ .

$\arccos (x) = \frac{π}{6}$

Step 3

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

$x = \cos (\frac{π}{6})$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$x = \frac{\sqrt{3}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{3}}{2}$

Decimal Form:

$x = 0.86602540 \dots$