Trigonometry Examples

$\sec (\frac{3 θ}{2}) = - 2$ , $[0, 2 π)$

Step 1

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

$\frac{3 θ}{2} = arcsec (- 2)$

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{3 θ}{2} = \frac{2 π}{3}$

Step 3

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

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

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 $\frac{2}{3} \cdot \frac{3 θ}{2}$ .

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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.

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

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 $n$

Step 9

Find the values of $n$ that produce a value within the interval $[0, 2 π)$ .

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Step 9.1

Plug in $0$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 9.1.1

Plug in $0$ for $n$ .

$\frac{4 π}{9} + \frac{4 π (0)}{3}$

Step 9.1.2

Simplify.

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Step 9.1.2.1

Simplify each term.

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Step 9.1.2.1.1

Cancel the common factor of $0$ and $3$ .

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Step 9.1.2.1.1.1

Factor $3$ out of $4 π (0)$ .

$\frac{4 π}{9} + \frac{3 (4 π \cdot (0))}{3}$

Step 9.1.2.1.1.2

Cancel the common factors.

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Step 9.1.2.1.1.2.1

Factor $3$ out of $3$ .

$\frac{4 π}{9} + \frac{3 (4 π \cdot (0))}{3 (1)}$

Step 9.1.2.1.1.2.2

Cancel the common factor.

$\frac{4 π}{9} + \frac{3 (4 π \cdot (0))}{3 \cdot 1}$

Step 9.1.2.1.1.2.3

Rewrite the expression.

$\frac{4 π}{9} + \frac{4 π \cdot (0)}{1}$

Step 9.1.2.1.1.2.4

Divide $4 π \cdot (0)$ by $1$ .

$\frac{4 π}{9} + 4 π \cdot (0)$

Step 9.1.2.1.2

Multiply $4 π (0)$ .

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Step 9.1.2.1.2.1

Multiply $0$ by $4$ .

$\frac{4 π}{9} + 0 π$

Step 9.1.2.1.2.2

Multiply $0$ by $π$ .

$\frac{4 π}{9} + 0$

Step 9.1.2.2

Add $\frac{4 π}{9}$ and $0$ .

$\frac{4 π}{9}$

Step 9.1.3

The interval $[0, 2 π)$ contains $\frac{4 π}{9}$ .

$θ = \frac{4 π}{9}$

Step 9.2

Plug in $0$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 9.2.1

Plug in $0$ for $n$ .

$\frac{8 π}{9} + \frac{4 π (0)}{3}$

Step 9.2.2

Simplify.

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Step 9.2.2.1

Simplify each term.

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Step 9.2.2.1.1

Cancel the common factor of $0$ and $3$ .

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Step 9.2.2.1.1.1

Factor $3$ out of $4 π (0)$ .

$\frac{8 π}{9} + \frac{3 (4 π \cdot (0))}{3}$

Step 9.2.2.1.1.2

Cancel the common factors.

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Step 9.2.2.1.1.2.1

Factor $3$ out of $3$ .

$\frac{8 π}{9} + \frac{3 (4 π \cdot (0))}{3 (1)}$

Step 9.2.2.1.1.2.2

Cancel the common factor.

$\frac{8 π}{9} + \frac{3 (4 π \cdot (0))}{3 \cdot 1}$

Step 9.2.2.1.1.2.3

Rewrite the expression.

$\frac{8 π}{9} + \frac{4 π \cdot (0)}{1}$

Step 9.2.2.1.1.2.4

Divide $4 π \cdot (0)$ by $1$ .

$\frac{8 π}{9} + 4 π \cdot (0)$

Step 9.2.2.1.2

Multiply $4 π (0)$ .

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Step 9.2.2.1.2.1

Multiply $0$ by $4$ .

$\frac{8 π}{9} + 0 π$

Step 9.2.2.1.2.2

Multiply $0$ by $π$ .

$\frac{8 π}{9} + 0$

Step 9.2.2.2

Add $\frac{8 π}{9}$ and $0$ .

$\frac{8 π}{9}$

Step 9.2.3

The interval $[0, 2 π)$ contains $\frac{8 π}{9}$ .

$θ = \frac{4 π}{9}, \frac{8 π}{9}$

Step 9.3

Plug in $1$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 9.3.1

Plug in $1$ for $n$ .

$\frac{4 π}{9} + \frac{4 π (1)}{3}$

Step 9.3.2

Simplify.

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Step 9.3.2.1

Multiply $4$ by $1$ .

$\frac{4 π}{9} + \frac{4 π}{3}$

Step 9.3.2.2

To write $\frac{4 π}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4 π}{9} + \frac{4 π}{3} \cdot \frac{3}{3}$

Step 9.3.2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.3.2.3.1

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

$\frac{4 π}{9} + \frac{4 π \cdot 3}{3 \cdot 3}$

Step 9.3.2.3.2

Multiply $3$ by $3$ .

$\frac{4 π}{9} + \frac{4 π \cdot 3}{9}$

Step 9.3.2.4

Combine the numerators over the common denominator.

$\frac{4 π + 4 π \cdot 3}{9}$

Step 9.3.2.5

Simplify the numerator.

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Step 9.3.2.5.1

Multiply $3$ by $4$ .

$\frac{4 π + 12 π}{9}$

Step 9.3.2.5.2

Add $4 π$ and $12 π$ .

$\frac{16 π}{9}$

Step 9.3.3

The interval $[0, 2 π)$ contains $\frac{16 π}{9}$ .

$θ = \frac{4 π}{9}, \frac{8 π}{9}, \frac{16 π}{9}$