Trigonometry Examples

$x^{2} - 5 y^{2} = 6$

Step 1

Solve for $y$ .

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Step 1.1

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

$- 5 y^{2} = 6 - x^{2}$

Step 1.2

Divide each term in $- 5 y^{2} = 6 - x^{2}$ by $- 5$ and simplify.

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Step 1.2.1

Divide each term in $- 5 y^{2} = 6 - x^{2}$ by $- 5$ .

$\frac{- 5 y^{2}}{- 5} = \frac{6}{- 5} + \frac{- x^{2}}{- 5}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 5 y^{2}}{- 5} = \frac{6}{- 5} + \frac{- x^{2}}{- 5}$

Step 1.2.2.1.2

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

$y^{2} = \frac{6}{- 5} + \frac{- x^{2}}{- 5}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$y^{2} = - \frac{6}{5} + \frac{- x^{2}}{- 5}$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

$y^{2} = - \frac{6}{5} + \frac{x^{2}}{5}$

Step 1.3

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

$y = \pm \sqrt{- \frac{6}{5} + \frac{x^{2}}{5}}$

Step 1.4

Simplify $\pm \sqrt{- \frac{6}{5} + \frac{x^{2}}{5}}$ .

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Step 1.4.1

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{- 6 + x^{2}}{5}}$

Step 1.4.2

Rewrite $\sqrt{\frac{- 6 + x^{2}}{5}}$ as $\frac{\sqrt{- 6 + x^{2}}}{\sqrt{5}}$ .

$y = \pm \frac{\sqrt{- 6 + x^{2}}}{\sqrt{5}}$

Step 1.4.3

Multiply $\frac{\sqrt{- 6 + x^{2}}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \pm \frac{\sqrt{- 6 + x^{2}}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 1.4.4

Combine and simplify the denominator.

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Step 1.4.4.1

Multiply $\frac{\sqrt{- 6 + x^{2}}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \pm \frac{\sqrt{- 6 + x^{2}} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 1.4.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$y = \pm \frac{\sqrt{- 6 + x^{2}} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 1.4.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$y = \pm \frac{\sqrt{- 6 + x^{2}} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 1.4.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{- 6 + x^{2}} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 1.4.4.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{- 6 + x^{2}} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 1.4.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{- 6 + x^{2}} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 1.4.4.6.2

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

$y = \pm \frac{\sqrt{- 6 + x^{2}} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 1.4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{\sqrt{- 6 + x^{2}} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.4.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.4.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{- 6 + x^{2}} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.4.4.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{- 6 + x^{2}} \sqrt{5}}{5^{1}}$

Step 1.4.4.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{- 6 + x^{2}} \sqrt{5}}{5}$

Step 1.4.5

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(- 6 + x^{2}) \cdot 5}}{5}$

Step 1.4.6

Reorder factors in $\pm \frac{\sqrt{(- 6 + x^{2}) \cdot 5}}{5}$ .

$y = \pm \frac{\sqrt{5 (- 6 + x^{2})}}{5}$

Step 1.5

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

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Step 1.5.1

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

$y = \frac{\sqrt{5 (- 6 + x^{2})}}{5}$

Step 1.5.2

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

$y = - \frac{\sqrt{5 (- 6 + x^{2})}}{5}$

Step 1.5.3

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

$y = \frac{\sqrt{5 (- 6 + x^{2})}}{5}$

$y = - \frac{\sqrt{5 (- 6 + x^{2})}}{5}$

$y = \frac{\sqrt{5 (- 6 + x^{2})}}{5}$

$y = - \frac{\sqrt{5 (- 6 + x^{2})}}{5}$

$y = \frac{\sqrt{5 (- 6 + x^{2})}}{5}$

$y = - \frac{\sqrt{5 (- 6 + x^{2})}}{5}$

Step 2

Find the domain to determine if the expression is continuous.

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Step 2.1

Set the radicand in $\sqrt{5 (- 6 + x^{2})}$ greater than or equal to $0$ to find where the expression is defined.

$5 (- 6 + x^{2}) \geq 0$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Simplify $5 (- 6 + x^{2})$ .

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Step 2.2.1.1

Apply the distributive property.

$5 \cdot - 6 + 5 x^{2} \geq 0$

Step 2.2.1.2

Multiply $5$ by $- 6$ .

$- 30 + 5 x^{2} \geq 0$

Step 2.2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = - \sqrt{6}, \sqrt{6}$

Step 2.2.3

Use each root to create test intervals.

$x < - \sqrt{6}$

$- \sqrt{6} < x < \sqrt{6}$

$x > \sqrt{6}$

Step 2.2.4

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 2.2.4.1

Test a value on the interval $x < - \sqrt{6}$ to see if it makes the inequality true.

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Step 2.2.4.1.1

Choose a value on the interval $x < - \sqrt{6}$ and see if this value makes the original inequality true.

$x = - 5$

Step 2.2.4.1.2

Replace $x$ with $- 5$ in the original inequality.

$5 (- 6 + {(- 5)}^{2}) \geq 0$

Step 2.2.4.1.3

The left side $95$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.2.4.2

Test a value on the interval $- \sqrt{6} < x < \sqrt{6}$ to see if it makes the inequality true.

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Step 2.2.4.2.1

Choose a value on the interval $- \sqrt{6} < x < \sqrt{6}$ and see if this value makes the original inequality true.

$x = 0$

Step 2.2.4.2.2

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

$5 (- 6 + {(0)}^{2}) \geq 0$

Step 2.2.4.2.3

The left side $- 30$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.2.4.3

Test a value on the interval $x > \sqrt{6}$ to see if it makes the inequality true.

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Step 2.2.4.3.1

Choose a value on the interval $x > \sqrt{6}$ and see if this value makes the original inequality true.

$x = 5$

Step 2.2.4.3.2

Replace $x$ with $5$ in the original inequality.

$5 (- 6 + {(5)}^{2}) \geq 0$

Step 2.2.4.3.3

The left side $95$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.2.4.4

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

$x < - \sqrt{6}$ True

$- \sqrt{6} < x < \sqrt{6}$ False

$x > \sqrt{6}$ True

$x < - \sqrt{6}$ True

$- \sqrt{6} < x < \sqrt{6}$ False

$x > \sqrt{6}$ True

Step 2.2.5

The solution consists of all of the true intervals.

$x \leq - \sqrt{6}$ or $x \geq \sqrt{6}$

Step 2.3

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

Interval Notation:

$(- \infty, - \sqrt{6}] \cup [\sqrt{6}, \infty)$

Set-Builder Notation:

${x | x \leq - \sqrt{6}, x \geq \sqrt{6}}$

Interval Notation:

$(- \infty, - \sqrt{6}] \cup [\sqrt{6}, \infty)$

Set-Builder Notation:

${x | x \leq - \sqrt{6}, x \geq \sqrt{6}}$

Step 3

Since the domain is not all real numbers, $\frac{\sqrt{5 (- 6 + x^{2})}}{5}, - \frac{\sqrt{5 (- 6 + x^{2})}}{5}$ is not continuous over all real numbers.

Not continuous

Step 4