Trigonometry Examples

$15 \arccos (\frac{y - 3 π}{4}) = 5 π$

Step 1

Divide each term in $15 \arccos (\frac{y - 3 π}{4}) = 5 π$ by $15$ and simplify.

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Step 1.1

Divide each term in $15 \arccos (\frac{y - 3 π}{4}) = 5 π$ by $15$ .

$\frac{15 \arccos (\frac{y - 3 π}{4})}{15} = \frac{5 π}{15}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $15$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{15 \arccos (\frac{y - 3 π}{4})}{15} = \frac{5 π}{15}$

Step 1.2.1.2

Divide $\arccos (\frac{y - 3 π}{4})$ by $1$ .

$\arccos (\frac{y - 3 π}{4}) = \frac{5 π}{15}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $5$ and $15$ .

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Step 1.3.1.1

Factor $5$ out of $5 π$ .

$\arccos (\frac{y - 3 π}{4}) = \frac{5 (π)}{15}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $5$ out of $15$ .

$\arccos (\frac{y - 3 π}{4}) = \frac{5 π}{5 \cdot 3}$

Step 1.3.1.2.2

Cancel the common factor.

$\arccos (\frac{y - 3 π}{4}) = \frac{5 π}{5 \cdot 3}$

Step 1.3.1.2.3

Rewrite the expression.

$\arccos (\frac{y - 3 π}{4}) = \frac{π}{3}$

Step 2

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

$\frac{y - 3 π}{4} = \cos (\frac{π}{3})$

Step 3

Simplify the left side.

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Step 3.1

Simplify $\frac{y - 3 π}{4}$ .

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Step 3.1.1

Split the fraction $\frac{y - 3 π}{4}$ into two fractions.

$\frac{y}{4} + \frac{- 3 π}{4} = \cos (\frac{π}{3})$

Step 3.1.2

Move the negative in front of the fraction.

$\frac{y}{4} - \frac{3 π}{4} = \cos (\frac{π}{3})$

Step 4

Simplify the right side.

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Step 4.1

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

$\frac{y}{4} - \frac{3 π}{4} = \frac{1}{2}$

Step 5

Add $\frac{3 π}{4}$ to both sides of the equation.

$\frac{y}{4} = \frac{1}{2} + \frac{3 π}{4}$

Step 6

Multiply both sides of the equation by $4$ .

$4 \frac{y}{4} = 4 (\frac{1}{2} + \frac{3 π}{4})$

Step 7

Simplify both sides of the equation.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Cancel the common factor of $4$ .

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Step 7.1.1.1

Cancel the common factor.

$4 \frac{y}{4} = 4 (\frac{1}{2} + \frac{3 π}{4})$

Step 7.1.1.2

Rewrite the expression.

$y = 4 (\frac{1}{2} + \frac{3 π}{4})$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $4 (\frac{1}{2} + \frac{3 π}{4})$ .

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Step 7.2.1.1

Apply the distributive property.

$y = 4 (\frac{1}{2}) + 4 \frac{3 π}{4}$

Step 7.2.1.2

Cancel the common factor of $2$ .

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Step 7.2.1.2.1

Factor $2$ out of $4$ .

$y = 2 (2) \frac{1}{2} + 4 \frac{3 π}{4}$

Step 7.2.1.2.2

Cancel the common factor.

$y = 2 \cdot 2 \frac{1}{2} + 4 \frac{3 π}{4}$

Step 7.2.1.2.3

Rewrite the expression.

$y = 2 + 4 \frac{3 π}{4}$

Step 7.2.1.3

Cancel the common factor of $4$ .

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Step 7.2.1.3.1

Cancel the common factor.

$y = 2 + 4 \frac{3 π}{4}$

Step 7.2.1.3.2

Rewrite the expression.

$y = 2 + 3 π$

Step 8

Add $\frac{3 π}{4}$ to both sides of the equation.

$\frac{y}{4} = \frac{1}{2} + \frac{3 π}{4}$

Step 9

Multiply both sides of the equation by $4$ .

$4 \frac{y}{4} = 4 (\frac{1}{2} + \frac{3 π}{4})$

Step 10

Simplify both sides of the equation.

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Step 10.1

Simplify the left side.

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Step 10.1.1

Cancel the common factor of $4$ .

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Step 10.1.1.1

Cancel the common factor.

$4 \frac{y}{4} = 4 (\frac{1}{2} + \frac{3 π}{4})$

Step 10.1.1.2

Rewrite the expression.

$y = 4 (\frac{1}{2} + \frac{3 π}{4})$

Step 10.2

Simplify the right side.

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Step 10.2.1

Simplify $4 (\frac{1}{2} + \frac{3 π}{4})$ .

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Step 10.2.1.1

Apply the distributive property.

$y = 4 (\frac{1}{2}) + 4 \frac{3 π}{4}$

Step 10.2.1.2

Cancel the common factor of $2$ .

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Step 10.2.1.2.1

Factor $2$ out of $4$ .

$y = 2 (2) \frac{1}{2} + 4 \frac{3 π}{4}$

Step 10.2.1.2.2

Cancel the common factor.

$y = 2 \cdot 2 \frac{1}{2} + 4 \frac{3 π}{4}$

Step 10.2.1.2.3

Rewrite the expression.

$y = 2 + 4 \frac{3 π}{4}$

Step 10.2.1.3

Cancel the common factor of $4$ .

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Step 10.2.1.3.1

Cancel the common factor.

$y = 2 + 4 \frac{3 π}{4}$

Step 10.2.1.3.2

Rewrite the expression.

$y = 2 + 3 π$

Step 11

The result can be shown in multiple forms.

Exact Form:

$y = 2 + 3 π$

Decimal Form:

$y = 11.42477796 \dots$