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

Add $2$ to 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

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

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

Step 2.2.3.2

Combine and simplify the denominator.

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Step 2.2.3.2.1

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

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

Step 2.2.3.2.2

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

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

Step 2.2.3.2.3

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

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

Step 2.2.3.2.4

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 2.2.3.2.5

Add $1$ and $1$ .

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

Step 2.2.3.2.6

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

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Step 2.2.3.2.6.1

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 2.2.3.2.6.2

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 2.2.3.2.6.3

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

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

Step 2.2.3.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.3.2.6.4.1

Cancel the common factor.

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

Step 2.2.3.2.6.4.2

Rewrite the expression.

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

Step 2.2.3.2.6.5

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 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.

$x = π - \frac{π}{3}$

Step 2.6

Simplify $π - \frac{π}{3}$ .

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Step 2.6.1

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

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

Step 2.6.2

Combine fractions.

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Step 2.6.2.1

Combine $π$ and $\frac{3}{3}$ .

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

Step 2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 - π}{3}$

Step 2.6.3

Simplify the numerator.

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Step 2.6.3.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π - π}{3}$

Step 2.6.3.2

Subtract $π$ from $3 π$ .

$x = \frac{2 π}{3}$

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

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

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

Step 3