Precalculus Examples

$y = - 3 \sin (\frac{1}{2} x + \frac{π}{5})$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - 3 \sin (\frac{1}{2} x + \frac{π}{5})$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 3 \sin (\frac{1}{2} x + \frac{π}{5}) = 0$ .

$- 3 \sin (\frac{1}{2} x + \frac{π}{5}) = 0$

Step 1.2.2

Divide each term in $- 3 \sin (\frac{1}{2} x + \frac{π}{5}) = 0$ by $- 3$ and simplify.

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Step 1.2.2.1

Divide each term in $- 3 \sin (\frac{1}{2} x + \frac{π}{5}) = 0$ by $- 3$ .

$\frac{- 3 \sin (\frac{1}{2} x + \frac{π}{5})}{- 3} = \frac{0}{- 3}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 \sin (\frac{1}{2} x + \frac{π}{5})}{- 3} = \frac{0}{- 3}$

Step 1.2.2.2.1.2

Divide $\sin (\frac{1}{2} x + \frac{π}{5})$ by $1$ .

$\sin (\frac{1}{2} x + \frac{π}{5}) = \frac{0}{- 3}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $- 3$ .

$\sin (\frac{1}{2} x + \frac{π}{5}) = 0$

Step 1.2.3

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

$\frac{1}{2} x + \frac{π}{5} = \arcsin (0)$

Step 1.2.4

Simplify the left side.

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Step 1.2.4.1

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{2} + \frac{π}{5} = \arcsin (0)$

Step 1.2.5

Simplify the right side.

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Step 1.2.5.1

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

$\frac{x}{2} + \frac{π}{5} = 0$

Step 1.2.6

Subtract $\frac{π}{5}$ from both sides of the equation.

$\frac{x}{2} = - \frac{π}{5}$

Step 1.2.7

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (- \frac{π}{5})$

Step 1.2.8

Simplify both sides of the equation.

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Step 1.2.8.1

Simplify the left side.

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Step 1.2.8.1.1

Cancel the common factor of $2$ .

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Step 1.2.8.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (- \frac{π}{5})$

Step 1.2.8.1.1.2

Rewrite the expression.

$x = 2 (- \frac{π}{5})$

Step 1.2.8.2

Simplify the right side.

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Step 1.2.8.2.1

Simplify $2 (- \frac{π}{5})$ .

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Step 1.2.8.2.1.1

Multiply $2 (- \frac{π}{5})$ .

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Step 1.2.8.2.1.1.1

Multiply $- 1$ by $2$ .

$x = - 2 \frac{π}{5}$

Step 1.2.8.2.1.1.2

Combine $- 2$ and $\frac{π}{5}$ .

$x = \frac{- 2 π}{5}$

Step 1.2.8.2.1.2

Move the negative in front of the fraction.

$x = - \frac{2 π}{5}$

Step 1.2.9

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.

$\frac{x}{2} + \frac{π}{5} = π - 0$

Step 1.2.10

Solve for $x$ .

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Step 1.2.10.1

Subtract $0$ from $π$ .

$\frac{x}{2} + \frac{π}{5} = π$

Step 1.2.10.2

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

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Step 1.2.10.2.1

Subtract $\frac{π}{5}$ from both sides of the equation.

$\frac{x}{2} = π - \frac{π}{5}$

Step 1.2.10.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x}{2} = π \cdot \frac{5}{5} - \frac{π}{5}$

Step 1.2.10.2.3

Combine $π$ and $\frac{5}{5}$ .

$\frac{x}{2} = \frac{π \cdot 5}{5} - \frac{π}{5}$

Step 1.2.10.2.4

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{π \cdot 5 - π}{5}$

Step 1.2.10.2.5

Simplify the numerator.

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Step 1.2.10.2.5.1

Move $5$ to the left of $π$ .

$\frac{x}{2} = \frac{5 \cdot π - π}{5}$

Step 1.2.10.2.5.2

Subtract $π$ from $5 π$ .

$\frac{x}{2} = \frac{4 π}{5}$

Step 1.2.10.3

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{4 π}{5}$

Step 1.2.10.4

Simplify both sides of the equation.

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Step 1.2.10.4.1

Simplify the left side.

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Step 1.2.10.4.1.1

Cancel the common factor of $2$ .

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Step 1.2.10.4.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{4 π}{5}$

Step 1.2.10.4.1.1.2

Rewrite the expression.

$x = 2 \frac{4 π}{5}$

Step 1.2.10.4.2

Simplify the right side.

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Step 1.2.10.4.2.1

Multiply $2 \frac{4 π}{5}$ .

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Step 1.2.10.4.2.1.1

Combine $2$ and $\frac{4 π}{5}$ .

$x = \frac{2 (4 π)}{5}$

Step 1.2.10.4.2.1.2

Multiply $4$ by $2$ .

$x = \frac{8 π}{5}$

Step 1.2.11

Find the period of $\sin (\frac{1}{2} x + \frac{π}{5})$ .

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Step 1.2.11.1

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

$\frac{2 π}{| b |}$

Step 1.2.11.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 1.2.11.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 1.2.11.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 1.2.11.5

Multiply $2$ by $2$ .

$4 π$

Step 1.2.12

Add $4 π$ to every negative angle to get positive angles.

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Step 1.2.12.1

Add $4 π$ to $- \frac{2 π}{5}$ to find the positive angle.

$- \frac{2 π}{5} + 4 π$

Step 1.2.12.2

To write $4 π$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$4 π \cdot \frac{5}{5} - \frac{2 π}{5}$

Step 1.2.12.3

Combine fractions.

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Step 1.2.12.3.1

Combine $4 π$ and $\frac{5}{5}$ .

$\frac{4 π \cdot 5}{5} - \frac{2 π}{5}$

Step 1.2.12.3.2

Combine the numerators over the common denominator.

$\frac{4 π \cdot 5 - 2 π}{5}$

Step 1.2.12.4

Simplify the numerator.

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Step 1.2.12.4.1

Multiply $5$ by $4$ .

$\frac{20 π - 2 π}{5}$

Step 1.2.12.4.2

Subtract $2 π$ from $20 π$ .

$\frac{18 π}{5}$

Step 1.2.12.5

List the new angles.

$x = \frac{18 π}{5}$

Step 1.2.13

The period of the $\sin (\frac{1}{2} x + \frac{π}{5})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = \frac{8 π}{5} + 4 π n, \frac{18 π}{5} + 4 π n$ , for any integer $n$

Step 1.2.14

Consolidate the answers.

$x = \frac{8 π}{5} + 2 π n$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{8 π}{5} + 2 π n, 0)$ , for any integer $n$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - 3 \sin (\frac{1}{2} \cdot (0) + \frac{π}{5})$

Step 2.2

Solve the equation.

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Step 2.2.1

Multiply $\frac{1}{2}$ by $0$ .

$y = - 3 \sin (\frac{1}{2} \cdot 0 + \frac{π}{5})$

Step 2.2.2

Remove parentheses.

$y = - 3 \sin (\frac{1}{2} \cdot (0) + \frac{π}{5})$

Step 2.2.3

Simplify $- 3 \sin (\frac{1}{2} \cdot (0) + \frac{π}{5})$ .

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Step 2.2.3.1

Multiply $\frac{1}{2}$ by $0$ .

$y = - 3 \sin (0 + \frac{π}{5})$

Step 2.2.3.2

Add $0$ and $\frac{π}{5}$ .

$y = - 3 \sin (\frac{π}{5})$

Step 2.2.3.3

Evaluate $\sin (\frac{π}{5})$ .

$y = - 3 \cdot 0.58778525$

Step 2.2.3.4

Multiply $- 3$ by $0.58778525$ .

$y = - 1.76335575$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1.76335575)$

Step 3

List the intersections.

x-intercept(s): $(\frac{8 π}{5} + 2 π n, 0)$ , for any integer $n$

y-intercept(s): $(0, - 1.76335575)$

Step 4