Trigonometry Examples

$\frac{3}{2} \arccos (\frac{y}{8}) = π$

Step 1

Multiply each term in $\frac{3}{2} \cdot \arccos (\frac{y}{8}) = π$ by $\frac{2}{3}$ to eliminate the fractions.

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Step 1.1

Multiply each term in $\frac{3}{2} \cdot \arccos (\frac{y}{8}) = π$ by $\frac{2}{3}$ .

$\frac{3}{2} \cdot \arccos (\frac{y}{8}) \frac{2}{3} = π \frac{2}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Combine $\frac{3}{2}$ and $\arccos (\frac{y}{8})$ .

$\frac{3 \arccos (\frac{y}{8})}{2} \cdot \frac{2}{3} = π \frac{2}{3}$

Step 1.2.2

Cancel the common factor of $3$ .

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Step 1.2.2.1

Factor $3$ out of $3 \arccos (\frac{y}{8})$ .

$\frac{3 (\arccos (\frac{y}{8}))}{2} \cdot \frac{2}{3} = π \frac{2}{3}$

Step 1.2.2.2

Cancel the common factor.

$\frac{3 \arccos (\frac{y}{8})}{2} \cdot \frac{2}{3} = π \frac{2}{3}$

Step 1.2.2.3

Rewrite the expression.

$\frac{\arccos (\frac{y}{8})}{2} \cdot 2 = π \frac{2}{3}$

Step 1.2.3

Cancel the common factor of $2$ .

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Step 1.2.3.1

Cancel the common factor.

$\frac{\arccos (\frac{y}{8})}{2} \cdot 2 = π \frac{2}{3}$

Step 1.2.3.2

Rewrite the expression.

$\arccos (\frac{y}{8}) = π \frac{2}{3}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Combine $π$ and $\frac{2}{3}$ .

$\arccos (\frac{y}{8}) = \frac{π \cdot 2}{3}$

Step 1.3.2

Move $2$ to the left of $π$ .

$\arccos (\frac{y}{8}) = \frac{2 π}{3}$

Step 2

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

$\frac{y}{8} = \cos (\frac{2 π}{3})$

Step 3

Simplify the right side.

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Step 3.1

Simplify $\cos (\frac{2 π}{3})$ .

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Step 3.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{y}{8} = - \cos (\frac{π}{3})$

Step 3.1.2

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

$\frac{y}{8} = - \frac{1}{2}$

Step 4

Multiply both sides of the equation by $8$ .

$8 \frac{y}{8} = 8 (- \frac{1}{2})$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $8$ .

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Step 5.1.1.1

Cancel the common factor.

$8 \frac{y}{8} = 8 (- \frac{1}{2})$

Step 5.1.1.2

Rewrite the expression.

$y = 8 (- \frac{1}{2})$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $8 (- \frac{1}{2})$ .

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y = 8 (\frac{- 1}{2})$

Step 5.2.1.1.2

Factor $2$ out of $8$ .

$y = 2 (4) \frac{- 1}{2}$

Step 5.2.1.1.3

Cancel the common factor.

$y = 2 \cdot 4 \frac{- 1}{2}$

Step 5.2.1.1.4

Rewrite the expression.

$y = 4 \cdot - 1$

Step 5.2.1.2

Multiply $4$ by $- 1$ .

$y = - 4$