Trigonometry Examples

$\frac{x^{2}}{64} + \frac{y^{2}}{80} = 1$

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 $1$ .

$\frac{x^{2}}{64} + \frac{y^{2}}{80} = 1$

Step 2

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

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

Step 3

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = 4 \sqrt{5}$

$b = 8$

$k = 0$

$h = 0$

Step 4

The center of an ellipse 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 ellipse by using the following formula.

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

Step 5.2

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

$\sqrt{{(4 \sqrt{5})}^{2} - {(8)}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the expression.

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Step 5.3.1.1

Apply the product rule to $4 \sqrt{5}$ .

$\sqrt{4^{2} {\sqrt{5}}^{2} - {(8)}^{2}}$

Step 5.3.1.2

Raise $4$ to the power of $2$ .

$\sqrt{16 {\sqrt{5}}^{2} - {(8)}^{2}}$

Step 5.3.2

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

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Step 5.3.2.1

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

$\sqrt{16 {(5^{\frac{1}{2}})}^{2} - {(8)}^{2}}$

Step 5.3.2.2

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

$\sqrt{16 \cdot 5^{\frac{1}{2} \cdot 2} - {(8)}^{2}}$

Step 5.3.2.3

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

$\sqrt{16 \cdot 5^{\frac{2}{2}} - {(8)}^{2}}$

Step 5.3.2.4

Cancel the common factor of $2$ .

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Step 5.3.2.4.1

Cancel the common factor.

$\sqrt{16 \cdot 5^{\frac{2}{2}} - {(8)}^{2}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{16 \cdot 5^{1} - {(8)}^{2}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{16 \cdot 5 - {(8)}^{2}}$

Step 5.3.3

Simplify the expression.

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Step 5.3.3.1

Multiply $16$ by $5$ .

$\sqrt{80 - {(8)}^{2}}$

Step 5.3.3.2

Raise $8$ to the power of $2$ .

$\sqrt{80 - 1 \cdot 64}$

Step 5.3.3.3

Multiply $- 1$ by $64$ .

$\sqrt{80 - 64}$

Step 5.3.3.4

Subtract $64$ from $80$ .

$\sqrt{16}$

Step 5.3.3.5

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2}}$

Step 5.3.3.6

Pull terms out from under the radical, assuming positive real numbers.

$4$

Step 6

Find the vertices.

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Step 6.1

The first vertex of an ellipse can be found by adding $a$ to $k$ .

$(h, k + a)$

Step 6.2

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

$(0, 0 + 4 \sqrt{5})$

Step 6.3

Simplify.

$(0, 4 \sqrt{5})$

Step 6.4

The second vertex of an ellipse can be found by subtracting $a$ from $k$ .

$(h, k - a)$

Step 6.5

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

$(0, 0 - (4 \sqrt{5}))$

Step 6.6

Simplify.

$(0, - 4 \sqrt{5})$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(0, 4 \sqrt{5})$

${Vertex}_{2}$ : $(0, - 4 \sqrt{5})$

${Vertex}_{1}$ : $(0, 4 \sqrt{5})$

${Vertex}_{2}$ : $(0, - 4 \sqrt{5})$

Step 7

Find the foci.

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Step 7.1

The first focus of an ellipse can be found by adding $c$ to $k$ .

$(h, k + c)$

Step 7.2

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

$(0, 0 + 4)$

Step 7.3

Simplify.

$(0, 4)$

Step 7.4

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Step 7.5

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

$(0, 0 - (4))$

Step 7.6

Simplify.

$(0, - 4)$

Step 7.7

Ellipses have two foci.

${Focus}_{1}$ : $(0, 4)$

${Focus}_{2}$ : $(0, - 4)$

${Focus}_{1}$ : $(0, 4)$

${Focus}_{2}$ : $(0, - 4)$

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.

$\frac{\sqrt{{(4 \sqrt{5})}^{2} - {(8)}^{2}}}{4 \sqrt{5}}$

Step 8.3

Simplify.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

Apply the product rule to $4 \sqrt{5}$ .

$\frac{\sqrt{4^{2} {\sqrt{5}}^{2} - 8^{2}}}{4 \sqrt{5}}$

Step 8.3.1.2

Raise $4$ to the power of $2$ .

$\frac{\sqrt{16 {\sqrt{5}}^{2} - 8^{2}}}{4 \sqrt{5}}$

Step 8.3.1.3

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

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Step 8.3.1.3.1

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

$\frac{\sqrt{16 {(5^{\frac{1}{2}})}^{2} - 8^{2}}}{4 \sqrt{5}}$

Step 8.3.1.3.2

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

$\frac{\sqrt{16 \cdot 5^{\frac{1}{2} \cdot 2} - 8^{2}}}{4 \sqrt{5}}$

Step 8.3.1.3.3

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

$\frac{\sqrt{16 \cdot 5^{\frac{2}{2}} - 8^{2}}}{4 \sqrt{5}}$

Step 8.3.1.3.4

Cancel the common factor of $2$ .

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Step 8.3.1.3.4.1

Cancel the common factor.

$\frac{\sqrt{16 \cdot 5^{\frac{2}{2}} - 8^{2}}}{4 \sqrt{5}}$

Step 8.3.1.3.4.2

Rewrite the expression.

$\frac{\sqrt{16 \cdot 5^{1} - 8^{2}}}{4 \sqrt{5}}$

Step 8.3.1.3.5

Evaluate the exponent.

$\frac{\sqrt{16 \cdot 5 - 8^{2}}}{4 \sqrt{5}}$

Step 8.3.1.4

Multiply $16$ by $5$ .

$\frac{\sqrt{80 - 8^{2}}}{4 \sqrt{5}}$

Step 8.3.1.5

Raise $8$ to the power of $2$ .

$\frac{\sqrt{80 - 1 \cdot 64}}{4 \sqrt{5}}$

Step 8.3.1.6

Multiply $- 1$ by $64$ .

$\frac{\sqrt{80 - 64}}{4 \sqrt{5}}$

Step 8.3.1.7

Subtract $64$ from $80$ .

$\frac{\sqrt{16}}{4 \sqrt{5}}$

Step 8.3.1.8

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2}}}{4 \sqrt{5}}$

Step 8.3.1.9

Pull terms out from under the radical, assuming positive real numbers.

$\frac{4}{4 \sqrt{5}}$

Step 8.3.2

Cancel the common factor of $4$ .

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Step 8.3.2.1

Cancel the common factor.

$\frac{4}{4 \sqrt{5}}$

Step 8.3.2.2

Rewrite the expression.

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

Step 8.3.3

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 8.3.4

Combine and simplify the denominator.

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Step 8.3.4.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 8.3.4.2

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

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

Step 8.3.4.3

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

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

Step 8.3.4.4

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

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

Step 8.3.4.5

Add $1$ and $1$ .

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

Step 8.3.4.6

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

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Step 8.3.4.6.1

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

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

Step 8.3.4.6.2

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

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

Step 8.3.4.6.3

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

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

Step 8.3.4.6.4

Cancel the common factor of $2$ .

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Step 8.3.4.6.4.1

Cancel the common factor.

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

Step 8.3.4.6.4.2

Rewrite the expression.

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

Step 8.3.4.6.5

Evaluate the exponent.

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

Step 9

These values represent the important values for graphing and analyzing an ellipse.

Center: $(0, 0)$

${Vertex}_{1}$ : $(0, 4 \sqrt{5})$

${Vertex}_{2}$ : $(0, - 4 \sqrt{5})$

${Focus}_{1}$ : $(0, 4)$

${Focus}_{2}$ : $(0, - 4)$

Eccentricity: $\frac{\sqrt{5}}{5}$

Step 10