Trigonometry Examples

$36 x^{2} - 9 y^{2} = 324$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Divide each term by $324$ to make the right side equal to one.

$\frac{36 x^{2}}{324} - \frac{9 y^{2}}{324} = \frac{324}{324}$

Step 1.2

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{x^{2}}{9} - \frac{y^{2}}{36} = 1$

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.

$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 3

Match the values in this hyperbola to those of the standard form. The variable $h$ represents the x-offset from the origin, $k$ represents the y-offset from origin, $a$ .

$a = 3$

$b = 6$

$k = 0$

$h = 0$

Step 4

The center of a hyperbola follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(0, 0)$

Step 5

Find $c$ , the distance from the center to a focus.

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Step 5.1

Find the distance from the center to a focus of the hyperbola by using the following formula.

$\sqrt{a^{2} + b^{2}}$

Step 5.2

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(3)}^{2} + {(6)}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Raise $3$ to the power of $2$ .

$\sqrt{9 + {(6)}^{2}}$

Step 5.3.2

Raise $6$ to the power of $2$ .

$\sqrt{9 + 36}$

Step 5.3.3

Add $9$ and $36$ .

$\sqrt{45}$

Step 5.3.4

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 5.3.4.1

Factor $9$ out of $45$ .

$\sqrt{9 (5)}$

Step 5.3.4.2

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

$\sqrt{3^{2} \cdot 5}$

Step 5.3.5

Pull terms out from under the radical.

$3 \sqrt{5}$

Step 6

Find the vertices.

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Step 6.1

The first vertex of a hyperbola can be found by adding $a$ to $h$ .

$(h + a, k)$

Step 6.2

Substitute the known values of $h$ , $a$ , and $k$ into the formula and simplify.

$(3, 0)$

Step 6.3

The second vertex of a hyperbola can be found by subtracting $a$ from $h$ .

$(h - a, k)$

Step 6.4

Substitute the known values of $h$ , $a$ , and $k$ into the formula and simplify.

$(- 3, 0)$

Step 6.5

The vertices of a hyperbola follow the form of $(h \pm a, k)$ . Hyperbolas have two vertices.

$(3, 0), (- 3, 0)$

Step 7

Find the foci.

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Step 7.1

The first focus of a hyperbola can be found by adding $c$ to $h$ .

$(h + c, k)$

Step 7.2

Substitute the known values of $h$ , $c$ , and $k$ into the formula and simplify.

$(3 \sqrt{5}, 0)$

Step 7.3

The second focus of a hyperbola can be found by subtracting $c$ from $h$ .

$(h - c, k)$

Step 7.4

Substitute the known values of $h$ , $c$ , and $k$ into the formula and simplify.

$(- 3 \sqrt{5}, 0)$

Step 7.5

The foci of a hyperbola follow the form of $(h \pm \sqrt{a^{2} + b^{2}}, k)$ . Hyperbolas have two foci.

$(3 \sqrt{5}, 0), (- 3 \sqrt{5}, 0)$

Step 8

Find the eccentricity.

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Step 8.1

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} + b^{2}}}{a}$

Step 8.2

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(3)}^{2} + {(6)}^{2}}}{3}$

Step 8.3

Simplify.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

Raise $3$ to the power of $2$ .

$\frac{\sqrt{9 + 6^{2}}}{3}$

Step 8.3.1.2

Raise $6$ to the power of $2$ .

$\frac{\sqrt{9 + 36}}{3}$

Step 8.3.1.3

Add $9$ and $36$ .

$\frac{\sqrt{45}}{3}$

Step 8.3.1.4

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 8.3.1.4.1

Factor $9$ out of $45$ .

$\frac{\sqrt{9 (5)}}{3}$

Step 8.3.1.4.2

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

$\frac{\sqrt{3^{2} \cdot 5}}{3}$

Step 8.3.1.5

Pull terms out from under the radical.

$\frac{3 \sqrt{5}}{3}$

Step 8.3.2

Cancel the common factor of $3$ .

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Step 8.3.2.1

Cancel the common factor.

$\frac{3 \sqrt{5}}{3}$

Step 8.3.2.2

Divide $\sqrt{5}$ by $1$ .

$\sqrt{5}$

Step 9

Find the focal parameter.

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Step 9.1

Find the value of the focal parameter of the hyperbola by using the following formula.

$\frac{b^{2}}{\sqrt{a^{2} + b^{2}}}$

Step 9.2

Substitute the values of $b$ and $\sqrt{a^{2} + b^{2}}$ in the formula.

$\frac{6^{2}}{3 \sqrt{5}}$

Step 9.3

Simplify.

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Step 9.3.1

Raise $6$ to the power of $2$ .

$\frac{36}{3 \sqrt{5}}$

Step 9.3.2

Cancel the common factor of $36$ and $3$ .

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Step 9.3.2.1

Factor $3$ out of $36$ .

$\frac{3 \cdot 12}{3 \sqrt{5}}$

Step 9.3.2.2

Cancel the common factors.

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Step 9.3.2.2.1

Factor $3$ out of $3 \sqrt{5}$ .

$\frac{3 \cdot 12}{3 (\sqrt{5})}$

Step 9.3.2.2.2

Cancel the common factor.

$\frac{3 \cdot 12}{3 \sqrt{5}}$

Step 9.3.2.2.3

Rewrite the expression.

$\frac{12}{\sqrt{5}}$

Step 9.3.3

Multiply $\frac{12}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{12}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 9.3.4

Combine and simplify the denominator.

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Step 9.3.4.1

Multiply $\frac{12}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{12 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 9.3.4.2

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

$\frac{12 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 9.3.4.3

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

$\frac{12 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 9.3.4.4

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

$\frac{12 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 9.3.4.5

Add $1$ and $1$ .

$\frac{12 \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 9.3.4.6

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

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Step 9.3.4.6.1

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

$\frac{12 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 9.3.4.6.2

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

$\frac{12 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 9.3.4.6.3

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

$\frac{12 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 9.3.4.6.4

Cancel the common factor of $2$ .

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Step 9.3.4.6.4.1

Cancel the common factor.

$\frac{12 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 9.3.4.6.4.2

Rewrite the expression.

$\frac{12 \sqrt{5}}{5^{1}}$

Step 9.3.4.6.5

Evaluate the exponent.

$\frac{12 \sqrt{5}}{5}$

Step 10

The asymptotes follow the form $y = \pm \frac{b (x - h)}{a} + k$ because this hyperbola opens left and right.

$y = \pm 2 \cdot x + 0$

Step 11

Add $2 \cdot x$ and $0$ .

$y = 2 x$

Step 12

Add $- 2 \cdot x$ and $0$ .

$y = - 2 x$

Step 13

This hyperbola has two asymptotes.

$y = 2 x, y = - 2 x$

Step 14

These values represent the important values for graphing and analyzing a hyperbola.

Center: $(0, 0)$

Vertices: $(3, 0), (- 3, 0)$

Foci: $(3 \sqrt{5}, 0), (- 3 \sqrt{5}, 0)$

Eccentricity: $\sqrt{5}$

Focal Parameter: $\frac{12 \sqrt{5}}{5}$

Asymptotes: $y = 2 x$ , $y = - 2 x$

Step 15