Precalculus Examples

$\csc (x) = 16 \sin (x)$

Step 1

Divide each term in $\csc (x) = 16 \sin (x)$ by $\csc (x)$ and simplify.

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Step 1.1

Divide each term in $\csc (x) = 16 \sin (x)$ by $\csc (x)$ .

$\frac{\csc (x)}{\csc (x)} = \frac{16 \sin (x)}{\csc (x)}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $\csc (x)$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{\csc (x)}{\csc (x)} = \frac{16 \sin (x)}{\csc (x)}$

Step 1.2.1.2

Rewrite the expression.

$1 = \frac{16 \sin (x)}{\csc (x)}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Separate fractions.

$1 = \frac{16}{1} \cdot \frac{\sin (x)}{\csc (x)}$

Step 1.3.2

Rewrite $\csc (x)$ in terms of sines and cosines.

$1 = \frac{16}{1} \cdot \frac{\sin (x)}{\frac{1}{\sin (x)}}$

Step 1.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$1 = \frac{16}{1} (\sin (x) \sin (x))$

Step 1.3.4

Simplify.

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Step 1.3.4.1

Raise $\sin (x)$ to the power of $1$ .

$1 = \frac{16}{1} (\sin^{1} (x) \sin (x))$

Step 1.3.4.2

Raise $\sin (x)$ to the power of $1$ .

$1 = \frac{16}{1} (\sin^{1} (x) \sin^{1} (x))$

Step 1.3.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 = \frac{16}{1} {\sin (x)}^{1 + 1}$

Step 1.3.4.4

Add $1$ and $1$ .

$1 = \frac{16}{1} \sin^{2} (x)$

Step 1.3.5

Divide $16$ by $1$ .

$1 = 16 \sin^{2} (x)$

Step 2

Rewrite the equation as $16 \sin^{2} (x) = 1$ .

$16 \sin^{2} (x) = 1$

Step 3

Divide each term in $16 \sin^{2} (x) = 1$ by $16$ and simplify.

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Step 3.1

Divide each term in $16 \sin^{2} (x) = 1$ by $16$ .

$\frac{16 \sin^{2} (x)}{16} = \frac{1}{16}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $16$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{16 \sin^{2} (x)}{16} = \frac{1}{16}$

Step 3.2.1.2

Divide $\sin^{2} (x)$ by $1$ .

$\sin^{2} (x) = \frac{1}{16}$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sin (x) = \pm \sqrt{\frac{1}{16}}$

Step 5

Simplify $\pm \sqrt{\frac{1}{16}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{1}{16}}$ as $\frac{\sqrt{1}}{\sqrt{16}}$ .

$\sin (x) = \pm \frac{\sqrt{1}}{\sqrt{16}}$

Step 5.2

Any root of $1$ is $1$ .

$\sin (x) = \pm \frac{1}{\sqrt{16}}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Rewrite $16$ as $4^{2}$ .

$\sin (x) = \pm \frac{1}{\sqrt{4^{2}}}$

Step 5.3.2

Pull terms out from under the radical, assuming positive real numbers.

$\sin (x) = \pm \frac{1}{4}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$\sin (x) = \frac{1}{4}$

Step 6.2

Next, use the negative value of the $\pm$ to find the second solution.

$\sin (x) = - \frac{1}{4}$

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sin (x) = \frac{1}{4}, - \frac{1}{4}$

Step 7

Set up each of the solutions to solve for $x$ .

$\sin (x) = \frac{1}{4}$

$\sin (x) = - \frac{1}{4}$

Step 8

Solve for $x$ in $\sin (x) = \frac{1}{4}$ .

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Step 8.1

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

$x = \arcsin (\frac{1}{4})$

Step 8.2

Simplify the right side.

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Step 8.2.1

Evaluate $\arcsin (\frac{1}{4})$ .

$x = 0.25268025$

Step 8.3

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.

$x = (3.14159265) - 0.25268025$

Step 8.4

Solve for $x$ .

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Step 8.4.1

Remove parentheses.

$x = 3.14159265 - 0.25268025$

Step 8.4.2

Remove parentheses.

$x = (3.14159265) - 0.25268025$

Step 8.4.3

Subtract $0.25268025$ from $3.14159265$ .

$x = 2.88891239$

Step 8.5

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

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Step 8.5.1

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

$\frac{2 π}{| b |}$

Step 8.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 8.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 8.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 8.6

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

$x = 0.25268025 + 2 π n, 2.88891239 + 2 π n$ , for any integer $n$

Step 9

Solve for $x$ in $\sin (x) = - \frac{1}{4}$ .

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Step 9.1

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

$x = \arcsin (- \frac{1}{4})$

Step 9.2

Simplify the right side.

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Step 9.2.1

Evaluate $\arcsin (- \frac{1}{4})$ .

$x = - 0.25268025$

Step 9.3

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.

$x = 2 (3.14159265) + 0.25268025 + 3.14159265$

Step 9.4

Simplify the expression to find the second solution.

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Step 9.4.1

Subtract $2 π$ from $2 (3.14159265) + 0.25268025 + 3.14159265$ .

$x = 2 (3.14159265) + 0.25268025 + 3.14159265 - 2 π$

Step 9.4.2

The resulting angle of $3.3942729$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.25268025 + 3.14159265$ .

$x = 3.3942729$

Step 9.5

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

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Step 9.5.1

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

$\frac{2 π}{| b |}$

Step 9.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 9.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 9.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 9.6

Add $2 π$ to every negative angle to get positive angles.

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Step 9.6.1

Add $2 π$ to $- 0.25268025$ to find the positive angle.

$- 0.25268025 + 2 π$

Step 9.6.2

Subtract $0.25268025$ from $2 π$ .

$6.03050505$

Step 9.6.3

List the new angles.

$x = 6.03050505$

Step 9.7

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

$x = 3.3942729 + 2 π n, 6.03050505 + 2 π n$ , for any integer $n$

Step 10

List all of the solutions.

$x = 0.25268025 + 2 π n, 2.88891239 + 2 π n, 3.3942729 + 2 π n, 6.03050505 + 2 π n$ , for any integer $n$

Step 11

Consolidate the solutions.

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Step 11.1

Consolidate $0.25268025 + 2 π n$ and $3.3942729 + 2 π n$ to $0.25268025 + π n$ .

$x = 0.25268025 + π n, 2.88891239 + 2 π n, 6.03050505 + 2 π n$ , for any integer $n$

Step 11.2

Consolidate $2.88891239 + 2 π n$ and $6.03050505 + 2 π n$ to $2.88891239 + π n$ .

$x = 0.25268025 + π n, 2.88891239 + π n$ , for any integer $n$