Precalculus Examples

$\cos (θ) = - \frac{\sqrt{3}}{2}$

Step 1

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $2$ .

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

Step 2

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $2$ .

$\cos (θ) \cdot 2$

Step 3

Move $2$ to the left of $\cos (θ)$ .

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

Step 4

Simplify $\cos (θ) \cdot \frac{2}{2}$ .

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Step 4.1

Divide $2$ by $2$ .

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

Step 4.2

Multiply $\cos (θ)$ by $1$ .

$\cos (θ) = - \frac{\sqrt{3}}{2}$

Step 5

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

$θ = \arccos (- \frac{\sqrt{3}}{2})$

Step 6

Simplify the right side.

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Step 6.1

The exact value of $\arccos (- \frac{\sqrt{3}}{2})$ is $\frac{5 π}{6}$ .

$θ = \frac{5 π}{6}$

Step 7

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.

$θ = 2 π - \frac{5 π}{6}$

Step 8

Simplify $2 π - \frac{5 π}{6}$ .

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Step 8.1

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

$θ = 2 π \cdot \frac{6}{6} - \frac{5 π}{6}$

Step 8.2

Combine fractions.

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Step 8.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$θ = \frac{2 π \cdot 6}{6} - \frac{5 π}{6}$

Step 8.2.2

Combine the numerators over the common denominator.

$θ = \frac{2 π \cdot 6 - 5 π}{6}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Multiply $6$ by $2$ .

$θ = \frac{12 π - 5 π}{6}$

Step 8.3.2

Subtract $5 π$ from $12 π$ .

$θ = \frac{7 π}{6}$

Step 9

Find the period of $\cos (θ)$ .

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Step 9.1

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

$\frac{2 π}{| b |}$

Step 9.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 9.4

Divide $2 π$ by $1$ .

$2 π$

Step 10

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

$θ = \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n$ , for any integer $n$