Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify the right side.

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Step 2.1

The exact value of $arcsec (- \sqrt{2})$ is $\frac{3 π}{4}$ .

$\frac{3 x}{2} = \frac{3 π}{4}$

Step 3

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \cdot \frac{3 π}{4}$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{2}{3} \cdot \frac{3 x}{2}$ .

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Step 4.1.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \cdot \frac{3 π}{4}$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{2}{3} \cdot \frac{3 π}{4}$

Step 4.1.1.2

Cancel the common factor of $3$ .

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Step 4.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{2}{3} \cdot \frac{3 π}{4}$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{2}{3} \cdot \frac{3 π}{4}$

Step 4.1.1.2.3

Rewrite the expression.

$x = \frac{2}{3} \cdot \frac{3 π}{4}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{2}{3} \cdot \frac{3 π}{4}$ .

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Step 4.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1

Factor $2$ out of $4$ .

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

Step 4.2.1.1.2

Cancel the common factor.

$x = \frac{2}{3} \cdot \frac{3 π}{2 \cdot 2}$

Step 4.2.1.1.3

Rewrite the expression.

$x = \frac{1}{3} \cdot \frac{3 π}{2}$

Step 4.2.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.2.1

Factor $3$ out of $3 π$ .

$x = \frac{1}{3} \cdot \frac{3 (π)}{2}$

Step 4.2.1.2.2

Cancel the common factor.

$x = \frac{1}{3} \cdot \frac{3 π}{2}$

Step 4.2.1.2.3

Rewrite the expression.

$x = \frac{π}{2}$

Step 5

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$\frac{3 x}{2} = 2 π - \frac{3 π}{4}$

Step 6

Solve for $x$ .

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Step 6.1

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} (2 π - \frac{3 π}{4})$

Step 6.2

Simplify both sides of the equation.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify $\frac{2}{3} \cdot \frac{3 x}{2}$ .

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Step 6.2.1.1.1

Cancel the common factor of $2$ .

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Step 6.2.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} (2 π - \frac{3 π}{4})$

Step 6.2.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{2}{3} (2 π - \frac{3 π}{4})$

Step 6.2.1.1.2

Cancel the common factor of $3$ .

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Step 6.2.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{2}{3} (2 π - \frac{3 π}{4})$

Step 6.2.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{2}{3} (2 π - \frac{3 π}{4})$

Step 6.2.1.1.2.3

Rewrite the expression.

$x = \frac{2}{3} (2 π - \frac{3 π}{4})$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $\frac{2}{3} (2 π - \frac{3 π}{4})$ .

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Step 6.2.2.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{2}{3} (2 π \cdot \frac{4}{4} - \frac{3 π}{4})$

Step 6.2.2.1.2

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2}{3} (\frac{2 π \cdot 4}{4} - \frac{3 π}{4})$

Step 6.2.2.1.3

Combine the numerators over the common denominator.

$x = \frac{2}{3} \cdot \frac{2 π \cdot 4 - 3 π}{4}$

Step 6.2.2.1.4

Cancel the common factor of $2$ .

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Step 6.2.2.1.4.1

Factor $2$ out of $4$ .

$x = \frac{2}{3} \cdot \frac{2 π \cdot 4 - 3 π}{2 (2)}$

Step 6.2.2.1.4.2

Cancel the common factor.

$x = \frac{2}{3} \cdot \frac{2 π \cdot 4 - 3 π}{2 \cdot 2}$

Step 6.2.2.1.4.3

Rewrite the expression.

$x = \frac{1}{3} \cdot \frac{2 π \cdot 4 - 3 π}{2}$

Step 6.2.2.1.5

Multiply $4$ by $2$ .

$x = \frac{1}{3} \cdot \frac{8 π - 3 π}{2}$

Step 6.2.2.1.6

Multiply $\frac{1}{3}$ by $\frac{8 π - 3 π}{2}$ .

$x = \frac{8 π - 3 π}{3 \cdot 2}$

Step 6.2.2.1.7

Multiply $3$ by $2$ .

$x = \frac{8 π - 3 π}{6}$

Step 6.2.2.1.8

Subtract $3 π$ from $8 π$ .

$x = \frac{5 π}{6}$

Step 7

Find the period of $\sec (\frac{3 x}{2})$ .

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

Replace $b$ with $\frac{3}{2}$ in the formula for period.

$\frac{2 π}{| \frac{3}{2} |}$

Step 7.3

$\frac{3}{2}$ is approximately $1.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3}{2}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{3}$

Step 7.5

Multiply $2 π \frac{2}{3}$ .

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Step 7.5.1

Combine $\frac{2}{3}$ and $2$ .

$\frac{2 \cdot 2}{3} π$

Step 7.5.2

Multiply $2$ by $2$ .

$\frac{4}{3} π$

Step 7.5.3

Combine $\frac{4}{3}$ and $π$ .

$\frac{4 π}{3}$

Step 8

The period of the $\sec (\frac{3 x}{2})$ function is $\frac{4 π}{3}$ so values will repeat every $\frac{4 π}{3}$ radians in both directions.

$x = \frac{π}{2} + \frac{4 π n}{3}, \frac{5 π}{6} + \frac{4 π n}{3}$ , for any integer $n$