Precalculus Examples

$- 2 \cos (3 θ) = 1$

Step 1

Divide each term in $- 2 \cos (3 θ) = 1$ by $- 2$ and simplify.

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Step 1.1

Divide each term in $- 2 \cos (3 θ) = 1$ by $- 2$ .

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Divide $\cos (3 θ)$ by $1$ .

$\cos (3 θ) = \frac{1}{- 2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$\cos (3 θ) = - \frac{1}{2}$

Step 2

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

$3 θ = \arccos (- \frac{1}{2})$

Step 3

Simplify the right side.

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Step 3.1

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

$3 θ = \frac{2 π}{3}$

Step 4

Divide each term in $3 θ = \frac{2 π}{3}$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 θ = \frac{2 π}{3}$ by $3$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{2 π}{3}}{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{2 π}{3} \cdot \frac{1}{3}$

Step 4.3.2

Multiply $\frac{2 π}{3} \cdot \frac{1}{3}$ .

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Step 4.3.2.1

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

$θ = \frac{2 π}{3 \cdot 3}$

Step 4.3.2.2

Multiply $3$ by $3$ .

$θ = \frac{2 π}{9}$

Step 5

The cosine 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.

$3 θ = 2 π - \frac{2 π}{3}$

Step 6

Solve for $θ$ .

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Step 6.1

Simplify.

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Step 6.1.1

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

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

Step 6.1.2

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 6.1.3

Combine the numerators over the common denominator.

$3 θ = \frac{2 π \cdot 3 - 2 π}{3}$

Step 6.1.4

Multiply $3$ by $2$ .

$3 θ = \frac{6 π - 2 π}{3}$

Step 6.1.5

Subtract $2 π$ from $6 π$ .

$3 θ = \frac{4 π}{3}$

Step 6.2

Divide each term in $3 θ = \frac{4 π}{3}$ by $3$ and simplify.

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Step 6.2.1

Divide each term in $3 θ = \frac{4 π}{3}$ by $3$ .

$\frac{3 θ}{3} = \frac{\frac{4 π}{3}}{3}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{3 θ}{3} = \frac{\frac{4 π}{3}}{3}$

Step 6.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{4 π}{3}}{3}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{4 π}{3} \cdot \frac{1}{3}$

Step 6.2.3.2

Multiply $\frac{4 π}{3} \cdot \frac{1}{3}$ .

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Step 6.2.3.2.1

Multiply $\frac{4 π}{3}$ by $\frac{1}{3}$ .

$θ = \frac{4 π}{3 \cdot 3}$

Step 6.2.3.2.2

Multiply $3$ by $3$ .

$θ = \frac{4 π}{9}$

Step 7

Find the period of $\cos (3 θ)$ .

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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 $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 8

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

$θ = \frac{2 π}{9} + \frac{2 π n}{3}, \frac{4 π}{9} + \frac{2 π n}{3}$ , for any integer $n$