Trigonometry Examples

sec (\frac{3 θ}{2}) = - 2

$\sec (\frac{3 θ}{2}) = - 2$

Step 1

Take the inverse secant of both sides of the equation to extract

θ

$θ$ from inside the secant.

\frac{3 θ}{2} = arcsec (- 2)

$\frac{3 θ}{2} = arcsec (- 2)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

arcsec (- 2)

$arcsec (- 2)$ is

\frac{2 π}{3}

$\frac{2 π}{3}$ .

\frac{3 θ}{2} = \frac{2 π}{3}

$\frac{3 θ}{2} = \frac{2 π}{3}$

\frac{3 θ}{2} = \frac{2 π}{3}

$\frac{3 θ}{2} = \frac{2 π}{3}$

Step 3

Multiply both sides of the equation by

\frac{2}{3}

$\frac{2}{3}$ .

\frac{2}{3} \cdot \frac{3 θ}{2} = \frac{2}{3} \cdot \frac{2 π}{3}

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

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{2}{3} \cdot \frac{3 θ}{2}

$\frac{2}{3} \cdot \frac{3 θ}{2}$ .

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Step 4.1.1.1

Cancel the common factor of

2

$2$ .

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Step 4.1.1.1.1

Cancel the common factor.

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

Step 4.1.1.1.2

Rewrite the expression.

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

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

$3 θ$ .

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

Step 4.1.1.2.2

Cancel the common factor.

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

Step 4.1.1.2.3

Rewrite the expression.

$θ = \frac{2}{3} \cdot \frac{2 π}{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply

$\frac{2}{3} \cdot \frac{2 π}{3}$ .

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Step 4.2.1.1

Multiply

$\frac{2}{3}$ by

$\frac{2 π}{3}$ .

$θ = \frac{2 (2 π)}{3 \cdot 3}$

Step 4.2.1.2

Multiply

$2$ by

$2$ .

$θ = \frac{4 π}{3 \cdot 3}$

Step 4.2.1.3

Multiply

$3$ by

$3$ .

$θ = \frac{4 π}{9}$

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.

$\frac{3 θ}{2} = 2 π - \frac{2 π}{3}$

Step 6

Solve for

$θ$ .

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Step 6.1

Multiply both sides of the equation by

$\frac{2}{3}$ .

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

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

$\frac{2}{3} \cdot \frac{3 θ}{2}$ .

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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.

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

Step 6.2.1.1.1.2

Rewrite the expression.

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

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

$3 θ$ .

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

Step 6.2.1.1.2.2

Cancel the common factor.

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

Step 6.2.1.1.2.3

Rewrite the expression.

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

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify

$\frac{2}{3} (2 π - \frac{2 π}{3})$ .

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Step 6.2.2.1.1

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

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

Step 6.2.2.1.2

Combine fractions.

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Step 6.2.2.1.2.1

Combine

$2 π$ and

$\frac{3}{3}$ .

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

Step 6.2.2.1.2.2

Combine the numerators over the common denominator.

$θ = \frac{2}{3} \cdot \frac{2 π \cdot 3 - 2 π}{3}$

Step 6.2.2.1.3

Simplify the numerator.

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Step 6.2.2.1.3.1

Multiply

$3$ by

$2$ .

$θ = \frac{2}{3} \cdot \frac{6 π - 2 π}{3}$

Step 6.2.2.1.3.2

Subtract

$2 π$ from

$6 π$ .

$θ = \frac{2}{3} \cdot \frac{4 π}{3}$

Step 6.2.2.1.4

Multiply

$\frac{2}{3} \cdot \frac{4 π}{3}$ .

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Step 6.2.2.1.4.1

Multiply

$\frac{2}{3}$ by

$\frac{4 π}{3}$ .

$θ = \frac{2 (4 π)}{3 \cdot 3}$

Step 6.2.2.1.4.2

Multiply

$4$ by

$2$ .

$θ = \frac{8 π}{3 \cdot 3}$

Step 6.2.2.1.4.3

Multiply

$3$ by

$3$ .

$θ = \frac{8 π}{9}$

Step 7

Find the period of

$\sec (\frac{3 θ}{2})$ .

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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

$\sec (\frac{3 θ}{2})$ function is

$\frac{4 π}{3}$ so values will repeat every

$\frac{4 π}{3}$ radians in both directions.

$θ = \frac{4 π}{9} + \frac{4 π n}{3}, \frac{8 π}{9} + \frac{4 π n}{3}$ , for any integer

$n$