Precalculus Examples

$4 \cos^{2} (x) - 4 \cos (x) + 1 = 0$

Step 1

Substitute $u$ for $\cos (x)$ .

$4 {(u)}^{2} - 4 u + 1 = 0$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $4 u^{2}$ as ${(2 u)}^{2}$ .

${(2 u)}^{2} - 4 u + 1 = 0$

Step 2.2

Rewrite $1$ as $1^{2}$ .

${(2 u)}^{2} - 4 u + 1^{2} = 0$

Step 2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 u = 2 \cdot (2 u) \cdot 1$

Step 2.4

Rewrite the polynomial.

${(2 u)}^{2} - 2 \cdot (2 u) \cdot 1 + 1^{2} = 0$

Step 2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 2 u$ and $b = 1$ .

${(2 u - 1)}^{2} = 0$

Step 3

Set the $2 u - 1$ equal to $0$ .

$2 u - 1 = 0$

Step 4

Solve for $u$ .

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Step 4.1

Add $1$ to both sides of the equation.

$2 u = 1$

Step 4.2

Divide each term in $2 u = 1$ by $2$ and simplify.

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Step 4.2.1

Divide each term in $2 u = 1$ by $2$ .

$\frac{2 u}{2} = \frac{1}{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{1}{2}$

Step 4.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{2}$

Step 5

Substitute $\cos (x)$ for $u$ .

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

Step 6

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

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

Step 7

Simplify the right side.

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Step 7.1

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

$x = \frac{π}{3}$

Step 8

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.

$x = 2 π - \frac{π}{3}$

Step 9

Simplify $2 π - \frac{π}{3}$ .

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Step 9.1

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

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

Step 9.2

Combine fractions.

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Step 9.2.1

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 9.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - π}{3}$

Step 9.3

Simplify the numerator.

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Step 9.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - π}{3}$

Step 9.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{3}$

Step 10

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

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Step 10.1

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

$\frac{2 π}{| b |}$

Step 10.2

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

$\frac{2 π}{| 1 |}$

Step 10.3

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

$\frac{2 π}{1}$

Step 10.4

Divide $2 π$ by $1$ .

$2 π$

Step 11

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

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$