Precalculus Examples

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

Step 1

Take the inverse cosine of both sides of the equation to extract

$x$ from inside the cosine.

$2 x = \arccos (- \frac{1}{2})$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

$\arccos (- \frac{1}{2})$ is

$\frac{2 π}{3}$ .

$2 x = \frac{2 π}{3}$

Step 3

Divide each term in

$2 x = \frac{2 π}{3}$ by

$2$ and simplify.

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Step 3.1

Divide each term in

$2 x = \frac{2 π}{3}$ by

$2$ .

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

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.

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

Step 3.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{2 π}{3}}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{2 π}{3} \cdot \frac{1}{2}$

Step 3.3.2

Cancel the common factor of

$2$ .

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Step 3.3.2.1

Factor

$2$ out of

$2 π$ .

$x = \frac{2 (π)}{3} \cdot \frac{1}{2}$

Step 3.3.2.2

Cancel the common factor.

$x = \frac{2 π}{3} \cdot \frac{1}{2}$

Step 3.3.2.3

Rewrite the expression.

$x = \frac{π}{3}$

Step 4

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.

$2 x = 2 π - \frac{2 π}{3}$

Step 5

Solve for

$x$ .

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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

$\frac{3}{3}$ .

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

Step 5.1.2

Combine

$2 π$ and

$\frac{3}{3}$ .

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

Step 5.1.3

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 3 - 2 π}{3}$

Step 5.1.4

Multiply

$3$ by

$2$ .

$2 x = \frac{6 π - 2 π}{3}$

Step 5.1.5

Subtract

$2 π$ from

$6 π$ .

$2 x = \frac{4 π}{3}$

Step 5.2

Divide each term in

$2 x = \frac{4 π}{3}$ by

$2$ and simplify.

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Step 5.2.1

Divide each term in

$2 x = \frac{4 π}{3}$ by

$2$ .

$\frac{2 x}{2} = \frac{\frac{4 π}{3}}{2}$

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.

$\frac{2 x}{2} = \frac{\frac{4 π}{3}}{2}$

Step 5.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{4 π}{3}}{2}$

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.

$x = \frac{4 π}{3} \cdot \frac{1}{2}$

Step 5.2.3.2

Cancel the common factor of

$2$ .

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Step 5.2.3.2.1

Factor

$2$ out of

$4 π$ .

$x = \frac{2 (2 π)}{3} \cdot \frac{1}{2}$

Step 5.2.3.2.2

Cancel the common factor.

$x = \frac{2 (2 π)}{3} \cdot \frac{1}{2}$

Step 5.2.3.2.3

Rewrite the expression.

$x = \frac{2 π}{3}$

Step 6

Find the period of

$\cos (2 x)$ .

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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

$\cos (2 x)$ function is

$π$ so values will repeat every

$π$ radians in both directions.

$x = \frac{π}{3} + π n, \frac{2 π}{3} + π n$ , for any integer

$n$