Trigonometry Examples

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

Step 1

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

$1$ .

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

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.

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

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,

$a$ .

$a = 1$

$b = 1$

$k = 0$

$h = 0$

Step 4

Find

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

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Step 4.1

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

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

Step 4.2

Substitute the values of

$a$ and

$b$ in the formula.

$\sqrt{{(1)}^{2} + {(1)}^{2}}$

Step 4.3

Simplify.

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Step 4.3.1

One to any power is one.

$\sqrt{1 + {(1)}^{2}}$

Step 4.3.2

One to any power is one.

$\sqrt{1 + 1}$

Step 4.3.3

Add

$1$ and

$1$ .

$\sqrt{2}$

Step 5

Find the foci.

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Step 5.1

The first focus of a hyperbola can be found by adding

$c$ to

$k$ .

$(h, k + c)$

Step 5.2

Substitute the known values of

$h$ ,

$c$ , and

$k$ into the formula and simplify.

$(0, \sqrt{2})$

Step 5.3

The second focus of a hyperbola can be found by subtracting

$c$ from

$k$ .

$(h, k - c)$

Step 5.4

Substitute the known values of

$h$ ,

$c$ , and

$k$ into the formula and simplify.

$(0, - \sqrt{2})$

Step 5.5

The foci of a hyperbola follow the form of

$(h, k \pm \sqrt{a^{2} + b^{2}})$ . Hyperbolas have two foci.

$(0, \sqrt{2}), (0, - \sqrt{2})$

Step 6