Trigonometry Examples

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

Step 1

Subtract $\cos (x)$ from both sides of the equation.

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

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 3

Simplify each side of the equation.

Tap for more steps...

Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 - 3 \cos^{2} (x)}$ as ${(3 - 3 \cos^{2} (x))}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify ${(- {(3 - 3 \cos^{2} (x))}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1

Apply the product rule to $- {(3 - 3 \cos^{2} (x))}^{\frac{1}{2}}$ .

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

Step 3.2.1.2

Raise $- 1$ to the power of $2$ .

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

Step 3.2.1.3

Multiply ${({(3 - 3 \cos^{2} (x))}^{\frac{1}{2}})}^{2}$ by $1$ .

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

Step 3.2.1.4

Multiply the exponents in ${({(3 - 3 \cos^{2} (x))}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.1.4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.4.2.1

Cancel the common factor.

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

Step 3.2.1.4.2.2

Rewrite the expression.

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

Step 3.2.1.5

Simplify.

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

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Simplify ${(- \cos (x))}^{2}$ .

Tap for more steps...

Step 3.3.1.1

Apply the product rule to $- \cos (x)$ .

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

Step 3.3.1.2

Raise $- 1$ to the power of $2$ .

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

Step 3.3.1.3

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

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

Step 4

Solve for $\cos (x)$ .

Tap for more steps...

Step 4.1

Move all terms containing $\cos (x)$ to the left side of the equation.

Tap for more steps...

Step 4.1.1

Subtract $\cos^{2} (x)$ from both sides of the equation.

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

Step 4.1.2

Subtract $\cos^{2} (x)$ from $- 3 \cos^{2} (x)$ .

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

Step 4.2

Subtract $3$ from both sides of the equation.

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

Step 4.3

Divide each term in $- 4 \cos^{2} (x) = - 3$ by $- 4$ and simplify.

Tap for more steps...

Step 4.3.1

Divide each term in $- 4 \cos^{2} (x) = - 3$ by $- 4$ .

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

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 4.3.2.1.1

Cancel the common factor.

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

Step 4.3.2.1.2

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

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

Step 4.3.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1

Dividing two negative values results in a positive value.

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

Step 4.4

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

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

Step 4.5

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

Tap for more steps...

Step 4.5.1

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

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

Step 4.5.2

Simplify the denominator.

Tap for more steps...

Step 4.5.2.1

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

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

Step 4.5.2.2

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

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

Step 4.6

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

Tap for more steps...

Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 5

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

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

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

Step 6

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

Tap for more steps...

Step 6.1

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

Simplify the right side.

Tap for more steps...

Step 6.2.1

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

$x = \frac{π}{6}$

Step 6.3

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.

$x = 2 π - \frac{π}{6}$

Step 6.4

Simplify $2 π - \frac{π}{6}$ .

Tap for more steps...

Step 6.4.1

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 6.4.2

Combine fractions.

Tap for more steps...

Step 6.4.2.1

Combine $2 π$ and $\frac{6}{6}$ .

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

Step 6.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - π}{6}$

Step 6.4.3

Simplify the numerator.

Tap for more steps...

Step 6.4.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - π}{6}$

Step 6.4.3.2

Subtract $π$ from $12 π$ .

$x = \frac{11 π}{6}$

Step 6.5

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

Tap for more steps...

Step 6.5.1

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

$\frac{2 π}{| b |}$

Step 6.5.2

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

$\frac{2 π}{| 1 |}$

Step 6.5.3

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

$\frac{2 π}{1}$

Step 6.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.6

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

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

Step 7

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

Tap for more steps...

Step 7.1

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

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

Step 7.2

Simplify the right side.

Tap for more steps...

Step 7.2.1

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

$x = \frac{5 π}{6}$

Step 7.3

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.

$x = 2 π - \frac{5 π}{6}$

Step 7.4

Simplify $2 π - \frac{5 π}{6}$ .

Tap for more steps...

Step 7.4.1

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

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

Step 7.4.2

Combine fractions.

Tap for more steps...

Step 7.4.2.1

Combine $2 π$ and $\frac{6}{6}$ .

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

Step 7.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - 5 π}{6}$

Step 7.4.3

Simplify the numerator.

Tap for more steps...

Step 7.4.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - 5 π}{6}$

Step 7.4.3.2

Subtract $5 π$ from $12 π$ .

$x = \frac{7 π}{6}$

Step 7.5

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

Tap for more steps...

Step 7.5.1

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

$\frac{2 π}{| b |}$

Step 7.5.2

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

$\frac{2 π}{| 1 |}$

Step 7.5.3

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

$\frac{2 π}{1}$

Step 7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.6

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 $n$

Step 8

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 $n$

Step 9

Consolidate the solutions.

Tap for more steps...

Step 9.1

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 $n$

Step 9.2

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 $n$

Step 10

Exclude the solutions that do not make $\cos (x) - \sqrt{3 - 3 \cos^{2} (x)} = 0$ true.

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