Trigonometry Examples

$\sec (\frac{5 x}{4}) = - 2$

Step 1

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

$\frac{5 x}{4} = arcsec (- 2)$

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{5 x}{4} = \frac{2 π}{3}$

Step 3

Multiply both sides of the equation by $\frac{4}{5}$ .

$\frac{4}{5} \cdot \frac{5 x}{4} = \frac{4}{5} \cdot \frac{2 π}{3}$

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 $\frac{4}{5} \cdot \frac{5 x}{4}$ .

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Step 4.1.1.1

Cancel the common factor of $4$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{4}{5} \cdot \frac{5 x}{4} = \frac{4}{5} \cdot \frac{2 π}{3}$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{4}{5} \cdot \frac{2 π}{3}$

Step 4.1.1.2

Cancel the common factor of $5$ .

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Step 4.1.1.2.1

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{4}{5} \cdot \frac{2 π}{3}$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{4}{5} \cdot \frac{2 π}{3}$

Step 4.1.1.2.3

Rewrite the expression.

$x = \frac{4}{5} \cdot \frac{2 π}{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $\frac{4}{5} \cdot \frac{2 π}{3}$ .

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Step 4.2.1.1

Multiply $\frac{4}{5}$ by $\frac{2 π}{3}$ .

$x = \frac{4 (2 π)}{5 \cdot 3}$

Step 4.2.1.2

Multiply $2$ by $4$ .

$x = \frac{8 π}{5 \cdot 3}$

Step 4.2.1.3

Multiply $5$ by $3$ .

$x = \frac{8 π}{15}$

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.

$\frac{5 x}{4} = 2 π - \frac{2 π}{3}$

Step 6

Solve for $x$ .

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Step 6.1

Multiply both sides of the equation by $\frac{4}{5}$ .

$\frac{4}{5} \cdot \frac{5 x}{4} = \frac{4}{5} (2 π - \frac{2 π}{3})$

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 $\frac{4}{5} \cdot \frac{5 x}{4}$ .

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Step 6.2.1.1.1

Cancel the common factor of $4$ .

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Step 6.2.1.1.1.1

Cancel the common factor.

$\frac{4}{5} \cdot \frac{5 x}{4} = \frac{4}{5} (2 π - \frac{2 π}{3})$

Step 6.2.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{4}{5} (2 π - \frac{2 π}{3})$

Step 6.2.1.1.2

Cancel the common factor of $5$ .

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Step 6.2.1.1.2.1

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{4}{5} (2 π - \frac{2 π}{3})$

Step 6.2.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{4}{5} (2 π - \frac{2 π}{3})$

Step 6.2.1.1.2.3

Rewrite the expression.

$x = \frac{4}{5} (2 π - \frac{2 π}{3})$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $\frac{4}{5} (2 π - \frac{2 π}{3})$ .

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Step 6.2.2.1.1

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

$x = \frac{4}{5} (2 π \cdot \frac{3}{3} - \frac{2 π}{3})$

Step 6.2.2.1.2

Combine fractions.

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Step 6.2.2.1.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$x = \frac{4}{5} (\frac{2 π \cdot 3}{3} - \frac{2 π}{3})$

Step 6.2.2.1.2.2

Combine the numerators over the common denominator.

$x = \frac{4}{5} \cdot \frac{2 π \cdot 3 - 2 π}{3}$

Step 6.2.2.1.3

Simplify the numerator.

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Step 6.2.2.1.3.1

Multiply $3$ by $2$ .

$x = \frac{4}{5} \cdot \frac{6 π - 2 π}{3}$

Step 6.2.2.1.3.2

Subtract $2 π$ from $6 π$ .

$x = \frac{4}{5} \cdot \frac{4 π}{3}$

Step 6.2.2.1.4

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

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Step 6.2.2.1.4.1

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

$x = \frac{4 (4 π)}{5 \cdot 3}$

Step 6.2.2.1.4.2

Multiply $4$ by $4$ .

$x = \frac{16 π}{5 \cdot 3}$

Step 6.2.2.1.4.3

Multiply $5$ by $3$ .

$x = \frac{16 π}{15}$

Step 7

Find the period of $\sec (\frac{5 x}{4})$ .

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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 $\frac{5}{4}$ in the formula for period.

$\frac{2 π}{| \frac{5}{4} |}$

Step 7.3

$\frac{5}{4}$ is approximately $1.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{5}{4}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{5}$

Step 7.5

Multiply $2 π \frac{4}{5}$ .

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Step 7.5.1

Combine $\frac{4}{5}$ and $2$ .

$\frac{4 \cdot 2}{5} π$

Step 7.5.2

Multiply $4$ by $2$ .

$\frac{8}{5} π$

Step 7.5.3

Combine $\frac{8}{5}$ and $π$ .

$\frac{8 π}{5}$

Step 8

The period of the $\sec (\frac{5 x}{4})$ function is $\frac{8 π}{5}$ so values will repeat every $\frac{8 π}{5}$ radians in both directions.

$x = \frac{8 π}{15} + \frac{8 π n}{5}, \frac{16 π}{15} + \frac{8 π n}{5}$ , for any integer $n$