Precalculus Examples

$2 \sin (5 x) + \sqrt{3} = 0$

Step 1

Subtract $\sqrt{3}$ from both sides of the equation.

$2 \sin (5 x) = - \sqrt{3}$

Step 2

Divide each term in $2 \sin (5 x) = - \sqrt{3}$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 \sin (5 x) = - \sqrt{3}$ by $2$ .

$\frac{2 \sin (5 x)}{2} = \frac{- \sqrt{3}}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 \sin (5 x)}{2} = \frac{- \sqrt{3}}{2}$

Step 2.2.1.2

Divide $\sin (5 x)$ by $1$ .

$\sin (5 x) = \frac{- \sqrt{3}}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$\sin (5 x) = - \frac{\sqrt{3}}{2}$

Step 3

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

$5 x = \arcsin (- \frac{\sqrt{3}}{2})$

Step 4

Simplify the right side.

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Step 4.1

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

$5 x = - \frac{π}{3}$

Step 5

Divide each term in $5 x = - \frac{π}{3}$ by $5$ and simplify.

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Step 5.1

Divide each term in $5 x = - \frac{π}{3}$ by $5$ .

$\frac{5 x}{5} = \frac{- \frac{π}{3}}{5}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $5$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- \frac{π}{3}}{5}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{3}}{5}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{3} \cdot \frac{1}{5}$

Step 5.3.2

Multiply $- \frac{π}{3} \cdot \frac{1}{5}$ .

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Step 5.3.2.1

Multiply $\frac{1}{5}$ by $\frac{π}{3}$ .

$x = - \frac{π}{5 \cdot 3}$

Step 5.3.2.2

Multiply $5$ by $3$ .

$x = - \frac{π}{15}$

Step 6

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$5 x = 2 π + \frac{π}{3} + π$

Step 7

Simplify the expression to find the second solution.

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Step 7.1

Subtract $2 π$ from $2 π + \frac{π}{3} + π$ .

$5 x = 2 π + \frac{π}{3} + π - 2 π$

Step 7.2

The resulting angle of $\frac{4 π}{3}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{3} + π$ .

$5 x = \frac{4 π}{3}$

Step 7.3

Divide each term in $5 x = \frac{4 π}{3}$ by $5$ and simplify.

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Step 7.3.1

Divide each term in $5 x = \frac{4 π}{3}$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{4 π}{3}}{5}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $5$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{4 π}{3}}{5}$

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{4 π}{3}}{5}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{4 π}{3} \cdot \frac{1}{5}$

Step 7.3.3.2

Multiply $\frac{4 π}{3} \cdot \frac{1}{5}$ .

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Step 7.3.3.2.1

Multiply $\frac{4 π}{3}$ by $\frac{1}{5}$ .

$x = \frac{4 π}{3 \cdot 5}$

Step 7.3.3.2.2

Multiply $3$ by $5$ .

$x = \frac{4 π}{15}$

Step 8

Find the period of $\sin (5 x)$ .

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

Replace $b$ with $5$ in the formula for period.

$\frac{2 π}{| 5 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{2 π}{5}$

Step 9

Add $\frac{2 π}{5}$ to every negative angle to get positive angles.

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Step 9.1

Add $\frac{2 π}{5}$ to $- \frac{π}{15}$ to find the positive angle.

$- \frac{π}{15} + \frac{2 π}{5}$

Step 9.2

To write $\frac{2 π}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 π}{5} \cdot \frac{3}{3} - \frac{π}{15}$

Step 9.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.3.1

Multiply $\frac{2 π}{5}$ by $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{5 \cdot 3} - \frac{π}{15}$

Step 9.3.2

Multiply $5$ by $3$ .

$\frac{2 π \cdot 3}{15} - \frac{π}{15}$

Step 9.4

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - π}{15}$

Step 9.5

Simplify the numerator.

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Step 9.5.1

Multiply $3$ by $2$ .

$\frac{6 π - π}{15}$

Step 9.5.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{15}$

Step 9.6

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

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Step 9.6.1

Factor $5$ out of $5 π$ .

$\frac{5 (π)}{15}$

Step 9.6.2

Cancel the common factors.

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Step 9.6.2.1

Factor $5$ out of $15$ .

$\frac{5 π}{5 \cdot 3}$

Step 9.6.2.2

Cancel the common factor.

$\frac{5 π}{5 \cdot 3}$

Step 9.6.2.3

Rewrite the expression.

$\frac{π}{3}$

Step 9.7

List the new angles.

$x = \frac{π}{3}$

Step 10

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

$x = \frac{4 π}{15} + \frac{2 π n}{5}, \frac{π}{3} + \frac{2 π n}{5}$ , for any integer $n$