Find All Complex Solutions cos(x)^2-3/4=0

$\cos^{2} (x) - \frac{3}{4} = 0$

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $4$ .

$\cos^{2} (x) \cdot \frac{4}{4} - \frac{3}{4} = 0 \cdot \frac{4}{4}$

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $4$ .

$\cos^{2} (x) \cdot 4$

Move $4$ to the left of $\cos^{2} (x)$ .

$\frac{4 \cos^{2} (x)}{4} - \frac{3}{4} = 0 \cdot \frac{4}{4}$

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $4$ .

$0 \cdot 4$

Multiply $0$ by $4$ .

$\frac{4 \cos^{2} (x)}{4} - \frac{3}{4} = \frac{0}{4}$

Simplify each term.

Divide $4$ by $4$ .

$\cos^{2} (x) \cdot 1 - \frac{3}{4} = 0 \cdot \frac{4}{4}$

Multiply $\cos^{2} (x)$ by $1$ .

$\cos^{2} (x) - \frac{3}{4} = 0 \cdot \frac{4}{4}$

Simplify $0 \cdot \frac{4}{4}$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\cos^{2} (x) - \frac{3}{4} = 0 \cdot \frac{4}{4}$

Rewrite the expression.

$\cos^{2} (x) - \frac{3}{4} = 0 \cdot 1$

Multiply $0$ by $1$ .

$\cos^{2} (x) - \frac{3}{4} = 0$

Add $\frac{3}{4}$ to both sides of the equation.

$\cos^{2} (x) = \frac{3}{4}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$\cos (x) = \pm \sqrt{\frac{3}{4}}$

Simplify $\pm \sqrt{\frac{3}{4}}$ .

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$\cos (x) = \pm \frac{\sqrt{3}}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

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

Next, use the negative value of the $\pm$ to find the second solution.

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

The complete solution is the result of both the positive and negative portions of the solution.

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

Set up each of the solutions to solve for $x$ .

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

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

Solve for $x$ in $\cos (x) = \frac{\sqrt{3}}{2}$ .

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

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

Simplify the right side.

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

$x = \frac{π}{6}$

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.

$x = 2 π - \frac{π}{6}$

Simplify $2 π - \frac{π}{6}$ .

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

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

Combine fractions.

Combine $2 π$ and $\frac{6}{6}$ .

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

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - π}{6}$

Simplify the numerator.

Multiply $6$ by $2$ .

$x = \frac{12 π - π}{6}$

Subtract $π$ from $12 π$ .

$x = \frac{11 π}{6}$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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 $n$

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

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

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

Simplify the right side.

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

$x = \frac{5 π}{6}$

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.

$x = 2 π - \frac{5 π}{6}$

Simplify $2 π - \frac{5 π}{6}$ .

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

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

Combine fractions.

Combine $2 π$ and $\frac{6}{6}$ .

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

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - 5 π}{6}$

Simplify the numerator.

Multiply $6$ by $2$ .

$x = \frac{12 π - 5 π}{6}$

Subtract $5 π$ from $12 π$ .

$x = \frac{7 π}{6}$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

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

List all of the solutions.

$x = \frac{π}{6} + 2 π n, \frac{11 π}{6} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n$ , for any integer $n$

Consolidate the solutions.

Consolidate $\frac{π}{6} + 2 π n$ and $\frac{7 π}{6} + 2 π n$ to $\frac{π}{6} + π n$ .

$x = \frac{π}{6} + π n, \frac{11 π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Consolidate $\frac{11 π}{6} + 2 π n$ and $\frac{5 π}{6} + 2 π n$ to $\frac{5 π}{6} + π n$ .

$x = \frac{π}{6} + π n, \frac{5 π}{6} + π n$ , for any integer $n$

Find All Complex Solutions cos(x)^2-3/4=0

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