Trigonometry Examples

$x^{2} - 9 y^{2} = 900$

Step 1

Subtract $x^{2}$ from both sides of the equation.

$- 9 y^{2} = 900 - x^{2}$

Step 2

Divide each term in $- 9 y^{2} = 900 - x^{2}$ by $- 9$ and simplify.

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Step 2.1

Divide each term in $- 9 y^{2} = 900 - x^{2}$ by $- 9$ .

$\frac{- 9 y^{2}}{- 9} = \frac{900}{- 9} + \frac{- x^{2}}{- 9}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 9$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 9 y^{2}}{- 9} = \frac{900}{- 9} + \frac{- x^{2}}{- 9}$

Step 2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{900}{- 9} + \frac{- x^{2}}{- 9}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Divide $900$ by $- 9$ .

$y^{2} = - 100 + \frac{- x^{2}}{- 9}$

Step 2.3.1.2

Dividing two negative values results in a positive value.

$y^{2} = - 100 + \frac{x^{2}}{9}$

Step 3

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

$y = \pm \sqrt{- 100 + \frac{x^{2}}{9}}$

Step 4

Simplify $\pm \sqrt{- 100 + \frac{x^{2}}{9}}$ .

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Step 4.1

Rewrite $9$ as $3^{2}$ .

$y = \pm \sqrt{- 100 + \frac{x^{2}}{3^{2}}}$

Step 4.2

Rewrite $\frac{x^{2}}{3^{2}}$ as ${(\frac{x}{3})}^{2}$ .

$y = \pm \sqrt{- 100 + {(\frac{x}{3})}^{2}}$

Step 4.3

Simplify the expression.

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Step 4.3.1

Rewrite $100$ as $10^{2}$ .

$y = \pm \sqrt{- 10^{2} + {(\frac{x}{3})}^{2}}$

Step 4.3.2

Reorder $- 10^{2}$ and ${(\frac{x}{3})}^{2}$ .

$y = \pm \sqrt{{(\frac{x}{3})}^{2} - 10^{2}}$

Step 4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{x}{3}$ and $b = 10$ .

$y = \pm \sqrt{(\frac{x}{3} + 10) (\frac{x}{3} - 10)}$

Step 4.5

To write $10$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{(\frac{x}{3} + 10 \cdot \frac{3}{3}) (\frac{x}{3} - 10)}$

Step 4.6

Combine $10$ and $\frac{3}{3}$ .

$y = \pm \sqrt{(\frac{x}{3} + \frac{10 \cdot 3}{3}) (\frac{x}{3} - 10)}$

Step 4.7

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x + 10 \cdot 3}{3} (\frac{x}{3} - 10)}$

Step 4.8

Multiply $10$ by $3$ .

$y = \pm \sqrt{\frac{x + 30}{3} (\frac{x}{3} - 10)}$

Step 4.9

To write $- 10$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{x + 30}{3} (\frac{x}{3} - 10 \cdot \frac{3}{3})}$

Step 4.10

Combine $- 10$ and $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{x + 30}{3} (\frac{x}{3} + \frac{- 10 \cdot 3}{3})}$

Step 4.11

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x + 30}{3} \cdot \frac{x - 10 \cdot 3}{3}}$

Step 4.12

Multiply $- 10$ by $3$ .

$y = \pm \sqrt{\frac{x + 30}{3} \cdot \frac{x - 30}{3}}$

Step 4.13

Multiply $\frac{x + 30}{3}$ by $\frac{x - 30}{3}$ .

$y = \pm \sqrt{\frac{(x + 30) (x - 30)}{3 \cdot 3}}$

Step 4.14

Multiply $3$ by $3$ .

$y = \pm \sqrt{\frac{(x + 30) (x - 30)}{9}}$

Step 4.15

Rewrite $\frac{(x + 30) (x - 30)}{9}$ as ${(\frac{1}{3})}^{2} ((x + 30) (x - 30))$ .

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Step 4.15.1

Factor the perfect power $1^{2}$ out of $(x + 30) (x - 30)$ .

$y = \pm \sqrt{\frac{1^{2} ((x + 30) (x - 30))}{9}}$

Step 4.15.2

Factor the perfect power $3^{2}$ out of $9$ .

$y = \pm \sqrt{\frac{1^{2} ((x + 30) (x - 30))}{3^{2} \cdot 1}}$

Step 4.15.3

Rearrange the fraction $\frac{1^{2} ((x + 30) (x - 30))}{3^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{1}{3})}^{2} ((x + 30) (x - 30))}$

Step 4.16

Pull terms out from under the radical.

$y = \pm \frac{1}{3} \sqrt{(x + 30) (x - 30)}$

Step 4.17

Combine $\frac{1}{3}$ and $\sqrt{(x + 30) (x - 30)}$ .

$y = \pm \frac{\sqrt{(x + 30) (x - 30)}}{3}$

Step 5

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

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Step 5.1

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

$y = \frac{\sqrt{(x + 30) (x - 30)}}{3}$

Step 5.2

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

$y = - \frac{\sqrt{(x + 30) (x - 30)}}{3}$

Step 5.3

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

$y = \frac{\sqrt{(x + 30) (x - 30)}}{3}$

$y = - \frac{\sqrt{(x + 30) (x - 30)}}{3}$

$y = \frac{\sqrt{(x + 30) (x - 30)}}{3}$

$y = - \frac{\sqrt{(x + 30) (x - 30)}}{3}$

Step 6

Set the radicand in $\sqrt{(x + 30) (x - 30)}$ greater than or equal to $0$ to find where the expression is defined.

$(x + 30) (x - 30) \geq 0$

Step 7

Solve for $x$ .

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Step 7.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 30 = 0$

$x - 30 = 0$

Step 7.2

Set $x + 30$ equal to $0$ and solve for $x$ .

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Step 7.2.1

Set $x + 30$ equal to $0$ .

$x + 30 = 0$

Step 7.2.2

Subtract $30$ from both sides of the equation.

$x = - 30$

Step 7.3

Set $x - 30$ equal to $0$ and solve for $x$ .

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Step 7.3.1

Set $x - 30$ equal to $0$ .

$x - 30 = 0$

Step 7.3.2

Add $30$ to both sides of the equation.

$x = 30$

Step 7.4

The final solution is all the values that make $(x + 30) (x - 30) \geq 0$ true.

$x = - 30, 30$

Step 7.5

Use each root to create test intervals.

$x < - 30$

$- 30 < x < 30$

$x > 30$

Step 7.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 7.6.1

Test a value on the interval $x < - 30$ to see if it makes the inequality true.

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Step 7.6.1.1

Choose a value on the interval $x < - 30$ and see if this value makes the original inequality true.

$x = - 32$

Step 7.6.1.2

Replace $x$ with $- 32$ in the original inequality.

$((- 32) + 30) ((- 32) - 30) \geq 0$

Step 7.6.1.3

The left side $124$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.6.2

Test a value on the interval $- 30 < x < 30$ to see if it makes the inequality true.

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Step 7.6.2.1

Choose a value on the interval $- 30 < x < 30$ and see if this value makes the original inequality true.

$x = 0$

Step 7.6.2.2

Replace $x$ with $0$ in the original inequality.

$((0) + 30) ((0) - 30) \geq 0$

Step 7.6.2.3

The left side $- 900$ is less than the right side $0$ , which means that the given statement is false.

False

Step 7.6.3

Test a value on the interval $x > 30$ to see if it makes the inequality true.

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Step 7.6.3.1

Choose a value on the interval $x > 30$ and see if this value makes the original inequality true.

$x = 32$

Step 7.6.3.2

Replace $x$ with $32$ in the original inequality.

$((32) + 30) ((32) - 30) \geq 0$

Step 7.6.3.3

The left side $124$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.6.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 30$ True

$- 30 < x < 30$ False

$x > 30$ True

$x < - 30$ True

$- 30 < x < 30$ False

$x > 30$ True

Step 7.7

The solution consists of all of the true intervals.

$x \leq - 30$ or $x \geq 30$

Step 8

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 30] \cup [30, \infty)$

Set-Builder Notation:

${x | x \leq - 30, x \geq 30}$

Step 9

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 10

Determine the domain and range.

Domain: $(- \infty, - 30] \cup [30, \infty), {x | x \leq - 30, x \geq 30}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 11