Precalculus Examples

$y = \sin (4 x + π)$

Step 1

Set $\sin (4 x + π)$ equal to $0$ .

$\sin (4 x + π) = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$4 x + π = \arcsin (0)$

Step 2.2

Simplify the right side.

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Step 2.2.1

The exact value of $\arcsin (0)$ is $0$ .

$4 x + π = 0$

Step 2.3

Subtract $π$ from both sides of the equation.

$4 x = - π$

Step 2.4

Divide each term in $4 x = - π$ by $4$ and simplify.

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Step 2.4.1

Divide each term in $4 x = - π$ by $4$ .

$\frac{4 x}{4} = \frac{- π}{4}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $4$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- π}{4}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- π}{4}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Move the negative in front of the fraction.

$x = - \frac{π}{4}$

Step 2.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$4 x + π = π - 0$

Step 2.6

Solve for $x$ .

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Step 2.6.1

Subtract $0$ from $π$ .

$4 x + π = π$

Step 2.6.2

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

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Step 2.6.2.1

Subtract $π$ from both sides of the equation.

$4 x = π - π$

Step 2.6.2.2

Subtract $π$ from $π$ .

$4 x = 0$

Step 2.6.3

Divide each term in $4 x = 0$ by $4$ and simplify.

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Step 2.6.3.1

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.6.3.2

Simplify the left side.

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Step 2.6.3.2.1

Cancel the common factor of $4$ .

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Step 2.6.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Step 2.6.3.3

Simplify the right side.

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Step 2.6.3.3.1

Divide $0$ by $4$ .

$x = 0$

Step 2.7

Find the period of $\sin (4 x + π)$ .

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Step 2.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 2.7.2

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Step 2.7.4

Cancel the common factor of $2$ and $4$ .

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Step 2.7.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 2.7.4.2

Cancel the common factors.

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Step 2.7.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 2.7.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 2.7.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 2.8

Add $\frac{π}{2}$ to every negative angle to get positive angles.

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Step 2.8.1

Add $\frac{π}{2}$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + \frac{π}{2}$

Step 2.8.2

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

$\frac{π}{2} \cdot \frac{2}{2} - \frac{π}{4}$

Step 2.8.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.8.3.1

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

$\frac{π \cdot 2}{2 \cdot 2} - \frac{π}{4}$

Step 2.8.3.2

Multiply $2$ by $2$ .

$\frac{π \cdot 2}{4} - \frac{π}{4}$

Step 2.8.4

Combine the numerators over the common denominator.

$\frac{π \cdot 2 - π}{4}$

Step 2.8.5

Simplify the numerator.

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Step 2.8.5.1

Move $2$ to the left of $π$ .

$\frac{2 \cdot π - π}{4}$

Step 2.8.5.2

Subtract $π$ from $2 π$ .

$\frac{π}{4}$

Step 2.8.6

List the new angles.

$x = \frac{π}{4}$

Step 2.9

The period of the $\sin (4 x + π)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{π n}{2}, \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$

Step 2.10

Consolidate the answers.

$x = \frac{π n}{4}$ , for any integer $n$

Step 3