Trigonometry Examples

$\frac{x^{2}}{169} + \frac{y^{2}}{25} = 1$

Step 1

Subtract $\frac{y^{2}}{25}$ from both sides of the equation.

$\frac{x^{2}}{169} = 1 - \frac{y^{2}}{25}$

Step 2

Multiply both sides of the equation by $169$ .

$169 \frac{x^{2}}{169} = 169 (1 - \frac{y^{2}}{25})$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $169$ .

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Step 3.1.1.1

Cancel the common factor.

$169 \frac{x^{2}}{169} = 169 (1 - \frac{y^{2}}{25})$

Step 3.1.1.2

Rewrite the expression.

$x^{2} = 169 (1 - \frac{y^{2}}{25})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $169 (1 - \frac{y^{2}}{25})$ .

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Step 3.2.1.1

Apply the distributive property.

$x^{2} = 169 \cdot 1 + 169 (- \frac{y^{2}}{25})$

Step 3.2.1.2

Multiply $169$ by $1$ .

$x^{2} = 169 + 169 (- \frac{y^{2}}{25})$

Step 3.2.1.3

Multiply $169 (- \frac{y^{2}}{25})$ .

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Step 3.2.1.3.1

Multiply $- 1$ by $169$ .

$x^{2} = 169 - 169 \frac{y^{2}}{25}$

Step 3.2.1.3.2

Combine $- 169$ and $\frac{y^{2}}{25}$ .

$x^{2} = 169 + \frac{- 169 y^{2}}{25}$

Step 3.2.1.4

Move the negative in front of the fraction.

$x^{2} = 169 - \frac{169 y^{2}}{25}$

Step 4

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

$x = \pm \sqrt{169 - \frac{169 y^{2}}{25}}$

Step 5

Simplify $\pm \sqrt{169 - \frac{169 y^{2}}{25}}$ .

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Step 5.1

Write the expression using exponents.

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Step 5.1.1

Rewrite $169$ as $13^{2}$ .

$x = \pm \sqrt{13^{2} - \frac{169 y^{2}}{25}}$

Step 5.1.2

Rewrite $\frac{169 y^{2}}{25}$ as ${(\frac{13 y}{5})}^{2}$ .

$x = \pm \sqrt{13^{2} - {(\frac{13 y}{5})}^{2}}$

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 13$ and $b = \frac{13 y}{5}$ .

$x = \pm \sqrt{(13 + \frac{13 y}{5}) (13 - \frac{13 y}{5})}$

Step 5.3

To write $13$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = \pm \sqrt{(13 \cdot \frac{5}{5} + \frac{13 y}{5}) (13 - \frac{13 y}{5})}$

Step 5.4

Combine $13$ and $\frac{5}{5}$ .

$x = \pm \sqrt{(\frac{13 \cdot 5}{5} + \frac{13 y}{5}) (13 - \frac{13 y}{5})}$

Step 5.5

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{13 \cdot 5 + 13 y}{5} (13 - \frac{13 y}{5})}$

Step 5.6

Factor $13$ out of $13 \cdot 5 + 13 y$ .

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Step 5.6.1

Factor $13$ out of $13 \cdot 5$ .

$x = \pm \sqrt{\frac{13 (5) + 13 y}{5} (13 - \frac{13 y}{5})}$

Step 5.6.2

Factor $13$ out of $13 y$ .

$x = \pm \sqrt{\frac{13 (5) + 13 (y)}{5} (13 - \frac{13 y}{5})}$

Step 5.6.3

Factor $13$ out of $13 (5) + 13 (y)$ .

$x = \pm \sqrt{\frac{13 (5 + y)}{5} (13 - \frac{13 y}{5})}$

Step 5.7

To write $13$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = \pm \sqrt{\frac{13 (5 + y)}{5} (13 \cdot \frac{5}{5} - \frac{13 y}{5})}$

Step 5.8

Combine $13$ and $\frac{5}{5}$ .

$x = \pm \sqrt{\frac{13 (5 + y)}{5} (\frac{13 \cdot 5}{5} - \frac{13 y}{5})}$

Step 5.9

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{13 (5 + y)}{5} \cdot \frac{13 \cdot 5 - 13 y}{5}}$

Step 5.10

Factor $13$ out of $13 \cdot 5 - 13 y$ .

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Step 5.10.1

Factor $13$ out of $13 \cdot 5$ .

$x = \pm \sqrt{\frac{13 (5 + y)}{5} \cdot \frac{13 (5) - 13 y}{5}}$

Step 5.10.2

Factor $13$ out of $- 13 y$ .

$x = \pm \sqrt{\frac{13 (5 + y)}{5} \cdot \frac{13 (5) + 13 (- y)}{5}}$

Step 5.10.3

Factor $13$ out of $13 (5) + 13 (- y)$ .

$x = \pm \sqrt{\frac{13 (5 + y)}{5} \cdot \frac{13 (5 - y)}{5}}$

Step 5.11

Multiply $\frac{13 (5 + y)}{5}$ by $\frac{13 (5 - y)}{5}$ .

$x = \pm \sqrt{\frac{13 (5 + y) (13 (5 - y))}{5 \cdot 5}}$

Step 5.12

Multiply.

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Step 5.12.1

Multiply $13$ by $13$ .

$x = \pm \sqrt{\frac{169 (5 + y) (5 - y)}{5 \cdot 5}}$

Step 5.12.2

Multiply $5$ by $5$ .

$x = \pm \sqrt{\frac{169 (5 + y) (5 - y)}{25}}$

Step 5.13

Rewrite $\frac{169 (5 + y) (5 - y)}{25}$ as ${(\frac{13}{5})}^{2} ((5 + y) (5 - y))$ .

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Step 5.13.1

Factor the perfect power $13^{2}$ out of $169 (5 + y) (5 - y)$ .

$x = \pm \sqrt{\frac{13^{2} ((5 + y) (5 - y))}{25}}$

Step 5.13.2

Factor the perfect power $5^{2}$ out of $25$ .

$x = \pm \sqrt{\frac{13^{2} ((5 + y) (5 - y))}{5^{2} \cdot 1}}$

Step 5.13.3

Rearrange the fraction $\frac{13^{2} ((5 + y) (5 - y))}{5^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{13}{5})}^{2} ((5 + y) (5 - y))}$

Step 5.14

Pull terms out from under the radical.

$x = \pm \frac{13}{5} \sqrt{(5 + y) (5 - y)}$

Step 5.15

Combine $\frac{13}{5}$ and $\sqrt{(5 + y) (5 - y)}$ .

$x = \pm \frac{13 \sqrt{(5 + y) (5 - y)}}{5}$

Step 6

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

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Step 6.1

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

$x = \frac{13 \sqrt{(5 + y) (5 - y)}}{5}$

Step 6.2

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

$x = - \frac{13 \sqrt{(5 + y) (5 - y)}}{5}$

Step 6.3

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

$x = \frac{13 \sqrt{(5 + y) (5 - y)}}{5}$

$x = - \frac{13 \sqrt{(5 + y) (5 - y)}}{5}$

$x = \frac{13 \sqrt{(5 + y) (5 - y)}}{5}$

$x = - \frac{13 \sqrt{(5 + y) (5 - y)}}{5}$