Precalculus Examples

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

Step 1

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

$\frac{x}{2} = \arccos (- \frac{\sqrt{3}}{2})$

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{x}{2} = \frac{5 π}{6}$

Step 3

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{5 π}{6}$

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

Cancel the common factor of $2$ .

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Step 4.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{5 π}{6}$

Step 4.1.1.2

Rewrite the expression.

$x = 2 \frac{5 π}{6}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Factor $2$ out of $6$ .

$x = 2 \frac{5 π}{2 (3)}$

Step 4.2.1.2

Cancel the common factor.

$x = 2 \frac{5 π}{2 \cdot 3}$

Step 4.2.1.3

Rewrite the expression.

$x = \frac{5 π}{3}$

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.

$\frac{x}{2} = 2 π - \frac{5 π}{6}$

Step 6

Solve for $x$ .

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Step 6.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (2 π - \frac{5 π}{6})$

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

Cancel the common factor of $2$ .

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Step 6.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (2 π - \frac{5 π}{6})$

Step 6.2.1.1.2

Rewrite the expression.

$x = 2 (2 π - \frac{5 π}{6})$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $2 (2 π - \frac{5 π}{6})$ .

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Step 6.2.2.1.1

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

$x = 2 (2 π \cdot \frac{6}{6} - \frac{5 π}{6})$

Step 6.2.2.1.2

Simplify terms.

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Step 6.2.2.1.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$x = 2 (\frac{2 π \cdot 6}{6} - \frac{5 π}{6})$

Step 6.2.2.1.2.2

Combine the numerators over the common denominator.

$x = 2 \frac{2 π \cdot 6 - 5 π}{6}$

Step 6.2.2.1.2.3

Cancel the common factor of $2$ .

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Step 6.2.2.1.2.3.1

Factor $2$ out of $6$ .

$x = 2 \frac{2 π \cdot 6 - 5 π}{2 (3)}$

Step 6.2.2.1.2.3.2

Cancel the common factor.

$x = 2 \frac{2 π \cdot 6 - 5 π}{2 \cdot 3}$

Step 6.2.2.1.2.3.3

Rewrite the expression.

$x = \frac{2 π \cdot 6 - 5 π}{3}$

Step 6.2.2.1.3

Simplify the numerator.

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Step 6.2.2.1.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - 5 π}{3}$

Step 6.2.2.1.3.2

Subtract $5 π$ from $12 π$ .

$x = \frac{7 π}{3}$

Step 7

Find the period of $\cos (\frac{x}{2})$ .

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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 $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 7.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 7.5

Multiply $2$ by $2$ .

$4 π$

Step 8

The period of the $\cos (\frac{x}{2})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = \frac{5 π}{3} + 4 π n, \frac{7 π}{3} + 4 π n$ , for any integer $n$