Trigonometry Examples

$\cos (2 θ) = \frac{\sqrt{3}}{2}$ , $[0, 2 π)$

Step 1

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

$2 θ = \arccos (\frac{\sqrt{3}}{2})$

Step 2

Simplify the right side.

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Step 2.1

The exact value of $\arccos (\frac{\sqrt{3}}{2})$ is $\frac{π}{6}$ .

$2 θ = \frac{π}{6}$

Step 3

Divide each term in $2 θ = \frac{π}{6}$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 θ = \frac{π}{6}$ by $2$ .

$\frac{2 θ}{2} = \frac{\frac{π}{6}}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 θ}{2} = \frac{\frac{π}{6}}{2}$

Step 3.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{π}{6}}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{π}{6} \cdot \frac{1}{2}$

Step 3.3.2

Multiply $\frac{π}{6} \cdot \frac{1}{2}$ .

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Step 3.3.2.1

Multiply $\frac{π}{6}$ by $\frac{1}{2}$ .

$θ = \frac{π}{6 \cdot 2}$

Step 3.3.2.2

Multiply $6$ by $2$ .

$θ = \frac{π}{12}$

Step 4

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$2 θ = 2 π - \frac{π}{6}$

Step 5

Solve for $θ$ .

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Step 5.1

Simplify.

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Step 5.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 5.1.2

Combine $2 π$ and $\frac{6}{6}$ .

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

Step 5.1.3

Combine the numerators over the common denominator.

$2 θ = \frac{2 π \cdot 6 - π}{6}$

Step 5.1.4

Multiply $6$ by $2$ .

$2 θ = \frac{12 π - π}{6}$

Step 5.1.5

Subtract $π$ from $12 π$ .

$2 θ = \frac{11 π}{6}$

Step 5.2

Divide each term in $2 θ = \frac{11 π}{6}$ by $2$ and simplify.

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Step 5.2.1

Divide each term in $2 θ = \frac{11 π}{6}$ by $2$ .

$\frac{2 θ}{2} = \frac{\frac{11 π}{6}}{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 θ}{2} = \frac{\frac{11 π}{6}}{2}$

Step 5.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{11 π}{6}}{2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{11 π}{6} \cdot \frac{1}{2}$

Step 5.2.3.2

Multiply $\frac{11 π}{6} \cdot \frac{1}{2}$ .

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Step 5.2.3.2.1

Multiply $\frac{11 π}{6}$ by $\frac{1}{2}$ .

$θ = \frac{11 π}{6 \cdot 2}$

Step 5.2.3.2.2

Multiply $6$ by $2$ .

$θ = \frac{11 π}{12}$

Step 6

Find the period of $\cos (2 θ)$ .

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 6.4.2

Divide $π$ by $1$ .

$π$

Step 7

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

$θ = \frac{π}{12} + π n, \frac{11 π}{12} + π n$ , for any integer $n$

Step 8

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

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Step 8.1

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

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Step 8.1.1

Plug in $0$ for $n$ .

$\frac{π}{12} + π (0)$

Step 8.1.2

Simplify.

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Step 8.1.2.1

Multiply $π$ by $0$ .

$\frac{π}{12} + 0$

Step 8.1.2.2

Add $\frac{π}{12}$ and $0$ .

$\frac{π}{12}$

Step 8.1.3

The interval $[0, 2 π)$ contains $\frac{π}{12}$ .

$θ = \frac{π}{12}$

Step 8.2

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

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Step 8.2.1

Plug in $0$ for $n$ .

$\frac{11 π}{12} + π (0)$

Step 8.2.2

Simplify.

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Step 8.2.2.1

Multiply $π$ by $0$ .

$\frac{11 π}{12} + 0$

Step 8.2.2.2

Add $\frac{11 π}{12}$ and $0$ .

$\frac{11 π}{12}$

Step 8.2.3

The interval $[0, 2 π)$ contains $\frac{11 π}{12}$ .

$θ = \frac{π}{12}, \frac{11 π}{12}$

Step 8.3

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

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Step 8.3.1

Plug in $1$ for $n$ .

$\frac{π}{12} + π (1)$

Step 8.3.2

Simplify.

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Step 8.3.2.1

Multiply $π$ by $1$ .

$\frac{π}{12} + π$

Step 8.3.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{π}{12} + π \cdot \frac{12}{12}$

Step 8.3.2.3

Combine fractions.

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Step 8.3.2.3.1

Combine $π$ and $\frac{12}{12}$ .

$\frac{π}{12} + \frac{π \cdot 12}{12}$

Step 8.3.2.3.2

Combine the numerators over the common denominator.

$\frac{π + π \cdot 12}{12}$

Step 8.3.2.4

Simplify the numerator.

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Step 8.3.2.4.1

Move $12$ to the left of $π$ .

$\frac{π + 12 \cdot π}{12}$

Step 8.3.2.4.2

Add $π$ and $12 π$ .

$\frac{13 π}{12}$

Step 8.3.3

The interval $[0, 2 π)$ contains $\frac{13 π}{12}$ .

$θ = \frac{π}{12}, \frac{11 π}{12}, \frac{13 π}{12}$

Step 8.4

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

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Step 8.4.1

Plug in $1$ for $n$ .

$\frac{11 π}{12} + π (1)$

Step 8.4.2

Simplify.

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Step 8.4.2.1

Multiply $π$ by $1$ .

$\frac{11 π}{12} + π$

Step 8.4.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{11 π}{12} + π \cdot \frac{12}{12}$

Step 8.4.2.3

Combine fractions.

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Step 8.4.2.3.1

Combine $π$ and $\frac{12}{12}$ .

$\frac{11 π}{12} + \frac{π \cdot 12}{12}$

Step 8.4.2.3.2

Combine the numerators over the common denominator.

$\frac{11 π + π \cdot 12}{12}$

Step 8.4.2.4

Simplify the numerator.

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Step 8.4.2.4.1

Move $12$ to the left of $π$ .

$\frac{11 π + 12 \cdot π}{12}$

Step 8.4.2.4.2

Add $11 π$ and $12 π$ .

$\frac{23 π}{12}$

Step 8.4.3

The interval $[0, 2 π)$ contains $\frac{23 π}{12}$ .

$θ = \frac{π}{12}, \frac{11 π}{12}, \frac{13 π}{12}, \frac{23 π}{12}$