Trigonometry Examples

$\frac{6}{5} \arccos (\frac{y}{6}) = π$

Step 1

Multiply each term in $\frac{6}{5} \cdot \arccos (\frac{y}{6}) = π$ by $\frac{5}{6}$ to eliminate the fractions.

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Step 1.1

Multiply each term in $\frac{6}{5} \cdot \arccos (\frac{y}{6}) = π$ by $\frac{5}{6}$ .

$\frac{6}{5} \cdot \arccos (\frac{y}{6}) \frac{5}{6} = π \frac{5}{6}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Combine $\frac{6}{5}$ and $\arccos (\frac{y}{6})$ .

$\frac{6 \arccos (\frac{y}{6})}{5} \cdot \frac{5}{6} = π \frac{5}{6}$

Step 1.2.2

Cancel the common factor of $6$ .

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Step 1.2.2.1

Factor $6$ out of $6 \arccos (\frac{y}{6})$ .

$\frac{6 (\arccos (\frac{y}{6}))}{5} \cdot \frac{5}{6} = π \frac{5}{6}$

Step 1.2.2.2

Cancel the common factor.

$\frac{6 \arccos (\frac{y}{6})}{5} \cdot \frac{5}{6} = π \frac{5}{6}$

Step 1.2.2.3

Rewrite the expression.

$\frac{\arccos (\frac{y}{6})}{5} \cdot 5 = π \frac{5}{6}$

Step 1.2.3

Cancel the common factor of $5$ .

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Step 1.2.3.1

Cancel the common factor.

$\frac{\arccos (\frac{y}{6})}{5} \cdot 5 = π \frac{5}{6}$

Step 1.2.3.2

Rewrite the expression.

$\arccos (\frac{y}{6}) = π \frac{5}{6}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Combine $π$ and $\frac{5}{6}$ .

$\arccos (\frac{y}{6}) = \frac{π \cdot 5}{6}$

Step 1.3.2

Move $5$ to the left of $π$ .

$\arccos (\frac{y}{6}) = \frac{5 π}{6}$

Step 2

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

$\frac{y}{6} = \cos (\frac{5 π}{6})$

Step 3

Simplify the right side.

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Step 3.1

Simplify $\cos (\frac{5 π}{6})$ .

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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.

$\frac{y}{6} = - \cos (\frac{π}{6})$

Step 3.1.2

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

$\frac{y}{6} = - \frac{\sqrt{3}}{2}$

Step 4

Multiply both sides of the equation by $6$ .

$6 \frac{y}{6} = 6 (- \frac{\sqrt{3}}{2})$

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 $6$ .

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Step 5.1.1.1

Cancel the common factor.

$6 \frac{y}{6} = 6 (- \frac{\sqrt{3}}{2})$

Step 5.1.1.2

Rewrite the expression.

$y = 6 (- \frac{\sqrt{3}}{2})$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $6 (- \frac{\sqrt{3}}{2})$ .

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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 $- \frac{\sqrt{3}}{2}$ into the numerator.

$y = 6 \frac{- \sqrt{3}}{2}$

Step 5.2.1.1.2

Factor $2$ out of $6$ .

$y = 2 (3) \frac{- \sqrt{3}}{2}$

Step 5.2.1.1.3

Cancel the common factor.

$y = 2 \cdot 3 \frac{- \sqrt{3}}{2}$

Step 5.2.1.1.4

Rewrite the expression.

$y = 3 (- \sqrt{3})$

Step 5.2.1.2

Multiply $- 1$ by $3$ .

$y = - 3 \sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = - 3 \sqrt{3}$

Decimal Form:

$y = - 5.19615242 \dots$