Trigonometry Examples

$\frac{y^{2}}{12} + \frac{x^{2}}{9} = 1$

Step 1

Subtract $\frac{y^{2}}{12}$ from both sides of the equation.

$\frac{x^{2}}{9} = 1 - \frac{y^{2}}{12}$

Step 2

Multiply both sides of the equation by $9$ .

$9 \frac{x^{2}}{9} = 9 (1 - \frac{y^{2}}{12})$

Step 3

Simplify both sides of the equation.

Tap for more steps...

Step 3.1

Simplify the left side.

Tap for more steps...

Step 3.1.1

Cancel the common factor of $9$ .

Tap for more steps...

Step 3.1.1.1

Cancel the common factor.

$9 \frac{x^{2}}{9} = 9 (1 - \frac{y^{2}}{12})$

Step 3.1.1.2

Rewrite the expression.

$x^{2} = 9 (1 - \frac{y^{2}}{12})$

Step 3.2

Simplify the right side.

Tap for more steps...

Step 3.2.1

Simplify $9 (1 - \frac{y^{2}}{12})$ .

Tap for more steps...

Step 3.2.1.1

Apply the distributive property.

$x^{2} = 9 \cdot 1 + 9 (- \frac{y^{2}}{12})$

Step 3.2.1.2

Multiply $9$ by $1$ .

$x^{2} = 9 + 9 (- \frac{y^{2}}{12})$

Step 3.2.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.3.1

Move the leading negative in $- \frac{y^{2}}{12}$ into the numerator.

$x^{2} = 9 + 9 \frac{- y^{2}}{12}$

Step 3.2.1.3.2

Factor $3$ out of $9$ .

$x^{2} = 9 + 3 (3) \frac{- y^{2}}{12}$

Step 3.2.1.3.3

Factor $3$ out of $12$ .

$x^{2} = 9 + 3 \cdot 3 \frac{- y^{2}}{3 \cdot 4}$

Step 3.2.1.3.4

Cancel the common factor.

$x^{2} = 9 + 3 \cdot 3 \frac{- y^{2}}{3 \cdot 4}$

Step 3.2.1.3.5

Rewrite the expression.

$x^{2} = 9 + 3 \frac{- y^{2}}{4}$

Step 3.2.1.4

Combine $3$ and $\frac{- y^{2}}{4}$ .

$x^{2} = 9 + \frac{3 (- y^{2})}{4}$

Step 3.2.1.5

Simplify the expression.

Tap for more steps...

Step 3.2.1.5.1

Multiply $- 1$ by $3$ .

$x^{2} = 9 + \frac{- 3 y^{2}}{4}$

Step 3.2.1.5.2

Move the negative in front of the fraction.

$x^{2} = 9 - \frac{3 y^{2}}{4}$

Step 4

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

$x = \pm \sqrt{9 - \frac{3 y^{2}}{4}}$

Step 5

Simplify $\pm \sqrt{9 - \frac{3 y^{2}}{4}}$ .

Tap for more steps...

Step 5.1

To write $9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \pm \sqrt{9 \cdot \frac{4}{4} - \frac{3 y^{2}}{4}}$

Step 5.2

Simplify terms.

Tap for more steps...

Step 5.2.1

Combine $9$ and $\frac{4}{4}$ .

$x = \pm \sqrt{\frac{9 \cdot 4}{4} - \frac{3 y^{2}}{4}}$

Step 5.2.2

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{9 \cdot 4 - 3 y^{2}}{4}}$

Step 5.3

Simplify the numerator.

Tap for more steps...

Step 5.3.1

Factor $3$ out of $9 \cdot 4 - 3 y^{2}$ .

Tap for more steps...

Step 5.3.1.1

Factor $3$ out of $9 \cdot 4$ .

$x = \pm \sqrt{\frac{3 (3 \cdot 4) - 3 y^{2}}{4}}$

Step 5.3.1.2

Factor $3$ out of $- 3 y^{2}$ .

$x = \pm \sqrt{\frac{3 (3 \cdot 4) + 3 (- y^{2})}{4}}$

Step 5.3.1.3

Factor $3$ out of $3 (3 \cdot 4) + 3 (- y^{2})$ .

$x = \pm \sqrt{\frac{3 (3 \cdot 4 - y^{2})}{4}}$

Step 5.3.2

Multiply $3$ by $4$ .

$x = \pm \sqrt{\frac{3 (12 - y^{2})}{4}}$

Step 5.4

Rewrite $\frac{3 (12 - y^{2})}{4}$ as ${(\frac{1}{2})}^{2} (3 (12 - y^{2}))$ .

Tap for more steps...

Step 5.4.1

Factor the perfect power $1^{2}$ out of $3 (12 - y^{2})$ .

$x = \pm \sqrt{\frac{1^{2} (3 (12 - y^{2}))}{4}}$

Step 5.4.2

Factor the perfect power $2^{2}$ out of $4$ .

$x = \pm \sqrt{\frac{1^{2} (3 (12 - y^{2}))}{2^{2} \cdot 1}}$

Step 5.4.3

Rearrange the fraction $\frac{1^{2} (3 (12 - y^{2}))}{2^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{1}{2})}^{2} (3 (12 - y^{2}))}$

Step 5.5

Pull terms out from under the radical.

$x = \pm \frac{1}{2} \sqrt{3 (12 - y^{2})}$

Step 5.6

Combine $\frac{1}{2}$ and $\sqrt{3 (12 - y^{2})}$ .

$x = \pm \frac{\sqrt{3 (12 - y^{2})}}{2}$

Step 6

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

Tap for more steps...

Step 6.1

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

$x = \frac{\sqrt{3 (12 - y^{2})}}{2}$

Step 6.2

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

$x = - \frac{\sqrt{3 (12 - y^{2})}}{2}$

Step 6.3

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

$x = \frac{\sqrt{3 (12 - y^{2})}}{2}$

$x = - \frac{\sqrt{3 (12 - y^{2})}}{2}$

$x = \frac{\sqrt{3 (12 - y^{2})}}{2}$

$x = - \frac{\sqrt{3 (12 - y^{2})}}{2}$