Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

$0 \cdot 4$

Step 5

Multiply $0$ by $4$ .

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

Step 6

Simplify each term.

Tap for more steps...

Step 6.1

Divide $4$ by $4$ .

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

Step 6.2

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

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

Step 7

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

Tap for more steps...

Step 7.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 7.1.1

Cancel the common factor.

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

Step 7.1.2

Rewrite the expression.

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

Step 7.2

Multiply $0$ by $1$ .

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

Step 8

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

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

Step 9

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

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

Step 10

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

Tap for more steps...

Step 10.1

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

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

Step 10.2

Simplify the denominator.

Tap for more steps...

Step 10.2.1

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

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

Step 10.2.2

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

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

Step 11

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

Tap for more steps...

Step 11.1

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

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

Step 11.2

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

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

Step 11.3

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

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

Step 12

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

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

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

Step 13

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

Tap for more steps...

Step 13.1

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

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

Step 13.2

Simplify the right side.

Tap for more steps...

Step 13.2.1

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

$x = \frac{π}{3}$

Step 13.3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = π - \frac{π}{3}$

Step 13.4

Simplify $π - \frac{π}{3}$ .

Tap for more steps...

Step 13.4.1

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

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

Step 13.4.2

Combine fractions.

Tap for more steps...

Step 13.4.2.1

Combine $π$ and $\frac{3}{3}$ .

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

Step 13.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 - π}{3}$

Step 13.4.3

Simplify the numerator.

Tap for more steps...

Step 13.4.3.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π - π}{3}$

Step 13.4.3.2

Subtract $π$ from $3 π$ .

$x = \frac{2 π}{3}$

Step 13.5

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

Tap for more steps...

Step 13.5.1

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

$\frac{2 π}{| b |}$

Step 13.5.2

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

$\frac{2 π}{| 1 |}$

Step 13.5.3

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

$\frac{2 π}{1}$

Step 13.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 13.6

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

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

Step 14

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

Tap for more steps...

Step 14.1

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

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

Step 14.2

Simplify the right side.

Tap for more steps...

Step 14.2.1

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

$x = - \frac{π}{3}$

Step 14.3

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{3} + π$

Step 14.4

Simplify the expression to find the second solution.

Tap for more steps...

Step 14.4.1

Subtract $2 π$ from $2 π + \frac{π}{3} + π$ .

$x = 2 π + \frac{π}{3} + π - 2 π$

Step 14.4.2

The resulting angle of $\frac{4 π}{3}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{3} + π$ .

$x = \frac{4 π}{3}$

Step 14.5

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

Tap for more steps...

Step 14.5.1

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

$\frac{2 π}{| b |}$

Step 14.5.2

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

$\frac{2 π}{| 1 |}$

Step 14.5.3

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

$\frac{2 π}{1}$

Step 14.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 14.6

Add $2 π$ to every negative angle to get positive angles.

Tap for more steps...

Step 14.6.1

Add $2 π$ to $- \frac{π}{3}$ to find the positive angle.

$- \frac{π}{3} + 2 π$

Step 14.6.2

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

$2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 14.6.3

Combine fractions.

Tap for more steps...

Step 14.6.3.1

Combine $2 π$ and $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 14.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - π}{3}$

Step 14.6.4

Simplify the numerator.

Tap for more steps...

Step 14.6.4.1

Multiply $3$ by $2$ .

$\frac{6 π - π}{3}$

Step 14.6.4.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{3}$

Step 14.6.5

List the new angles.

$x = \frac{5 π}{3}$

Step 14.7

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

$x = \frac{4 π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$

Step 15

List all of the solutions.

$x = \frac{π}{3} + 2 π n, \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$

Step 16

Consolidate the solutions.

Tap for more steps...

Step 16.1

Consolidate $\frac{π}{3} + 2 π n$ and $\frac{4 π}{3} + 2 π n$ to $\frac{π}{3} + π n$ .

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

Step 16.2

Consolidate $\frac{2 π}{3} + 2 π n$ and $\frac{5 π}{3} + 2 π n$ to $\frac{2 π}{3} + π n$ .

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