Precalculus Examples

$\cos (2 x) - \sqrt{3} \cos (x) = - 1$

Step 1

Add $1$ to both sides of the equation.

$\cos (2 x) - \sqrt{3} \cos (x) + 1 = 0$

Step 2

Simplify the left side of the equation.

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Step 2.1

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

$2 \cos^{2} (x) - 1 - \sqrt{3} \cos (x) + 1 = 0$

Step 2.2

Combine the opposite terms in $2 \cos^{2} (x) - 1 - \sqrt{3} \cos (x) + 1$ .

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Step 2.2.1

Add $- 1$ and $1$ .

$2 \cos^{2} (x) - \sqrt{3} \cos (x) + 0 = 0$

Step 2.2.2

Add $2 \cos^{2} (x) - \sqrt{3} \cos (x)$ and $0$ .

$2 \cos^{2} (x) - \sqrt{3} \cos (x) = 0$

Step 3

Factor $\cos (x)$ out of $2 \cos^{2} (x) - \sqrt{3} \cos (x)$ .

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Step 3.1

Factor $\cos (x)$ out of $2 \cos^{2} (x)$ .

$\cos (x) (2 \cos (x)) - \sqrt{3} \cos (x) = 0$

Step 3.2

Factor $\cos (x)$ out of $- \sqrt{3} \cos (x)$ .

$\cos (x) (2 \cos (x)) + \cos (x) (- \sqrt{3}) = 0$

Step 3.3

Factor $\cos (x)$ out of $\cos (x) (2 \cos (x)) + \cos (x) (- \sqrt{3})$ .

$\cos (x) (2 \cos (x) - \sqrt{3}) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cos (x) = 0$

$2 \cos (x) - \sqrt{3} = 0$

Step 5

Set $\cos (x)$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $\cos (x)$ equal to $0$ .

$\cos (x) = 0$

Step 5.2

Solve $\cos (x) = 0$ for $x$ .

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Step 5.2.1

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

$x = \arccos (0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 5.2.3

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.

$x = 2 π - \frac{π}{2}$

Step 5.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 5.2.4.1

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

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

Step 5.2.4.2

Combine fractions.

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Step 5.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 5.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 5.2.4.3

Simplify the numerator.

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Step 5.2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 5.2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 5.2.5

Find the period of $\cos (x)$ .

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Step 5.2.5.1

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

$\frac{2 π}{| b |}$

Step 5.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.5.3

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

$\frac{2 π}{1}$

Step 5.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.6

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

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

Step 6

Set $2 \cos (x) - \sqrt{3}$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $2 \cos (x) - \sqrt{3}$ equal to $0$ .

$2 \cos (x) - \sqrt{3} = 0$

Step 6.2

Solve $2 \cos (x) - \sqrt{3} = 0$ for $x$ .

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Step 6.2.1

Add $\sqrt{3}$ to both sides of the equation.

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

Step 6.2.2

Divide each term in $2 \cos (x) = \sqrt{3}$ by $2$ and simplify.

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Step 6.2.2.1

Divide each term in $2 \cos (x) = \sqrt{3}$ by $2$ .

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

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.2.1.2

Divide $\cos (x)$ by $1$ .

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

Step 6.2.3

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

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

Step 6.2.4

Simplify the right side.

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Step 6.2.4.1

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

$x = \frac{π}{6}$

Step 6.2.5

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.

$x = 2 π - \frac{π}{6}$

Step 6.2.6

Simplify $2 π - \frac{π}{6}$ .

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Step 6.2.6.1

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

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

Step 6.2.6.2

Combine fractions.

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Step 6.2.6.2.1

Combine $2 π$ and $\frac{6}{6}$ .

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

Step 6.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - π}{6}$

Step 6.2.6.3

Simplify the numerator.

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Step 6.2.6.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - π}{6}$

Step 6.2.6.3.2

Subtract $π$ from $12 π$ .

$x = \frac{11 π}{6}$

Step 6.2.7

Find the period of $\cos (x)$ .

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Step 6.2.7.1

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

$\frac{2 π}{| b |}$

Step 6.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 6.2.7.3

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

$\frac{2 π}{1}$

Step 6.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.2.8

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

$x = \frac{π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Step 7

The final solution is all the values that make $\cos (x) (2 \cos (x) - \sqrt{3}) = 0$ true.

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n, \frac{π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Step 8

Consolidate $\frac{π}{2} + 2 π n$ and $\frac{3 π}{2} + 2 π n$ to $\frac{π}{2} + π n$ .

$x = \frac{π}{2} + π n, \frac{π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$