Precalculus Examples

$f (x) = \sqrt{3} \csc (x) + 2$

Step 1

Set $\sqrt{3} \csc (x) + 2$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

Subtract $2$ from both sides of the equation.

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $\sqrt{3}$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $\csc (x)$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

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

Step 2.2.3.2

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\csc (x) = - (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 2.2.3.3

Combine and simplify the denominator.

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Step 2.2.3.3.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 2.2.3.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$\csc (x) = - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.2.3.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\csc (x) = - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.2.3.3.4

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

$\csc (x) = - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.2.3.3.5

Add $1$ and $1$ .

$\csc (x) = - \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.2.3.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.2.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\csc (x) = - \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.2.3.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\csc (x) = - \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.2.3.3.6.3

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

$\csc (x) = - \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.2.3.3.6.4

Cancel the common factor of $2$ .

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Step 2.2.3.3.6.4.1

Cancel the common factor.

$\csc (x) = - \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.2.3.3.6.4.2

Rewrite the expression.

$\csc (x) = - \frac{2 \sqrt{3}}{3^{1}}$

Step 2.2.3.3.6.5

Evaluate the exponent.

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

Step 2.3

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

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

Step 2.4

Simplify the right side.

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Step 2.4.1

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

$x = - \frac{π}{3}$

Step 2.5

The cosecant 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.

$x = 2 π + \frac{π}{3} + π$

Step 2.6

Simplify the expression to find the second solution.

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Step 2.6.1

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

$x = 2 π + \frac{π}{3} + π - 2 π$

Step 2.6.2

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

$x = \frac{4 π}{3}$

Step 2.7

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

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 2.7.3

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

$\frac{2 π}{1}$

Step 2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.8

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

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Step 2.8.1

Add $2 π$ to $- \frac{π}{3}$ to find the positive angle.

$- \frac{π}{3} + 2 π$

Step 2.8.2

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

$2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 2.8.3

Combine fractions.

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Step 2.8.3.1

Combine $2 π$ and $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 2.8.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - π}{3}$

Step 2.8.4

Simplify the numerator.

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Step 2.8.4.1

Multiply $3$ by $2$ .

$\frac{6 π - π}{3}$

Step 2.8.4.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{3}$

Step 2.8.5

List the new angles.

$x = \frac{5 π}{3}$

Step 2.9

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

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

Step 3