Trigonometry Examples

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

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$\frac{x^{2}}{25} - {(0)}^{2} = 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $\frac{x^{2}}{25} - {(0)}^{2}$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{x^{2}}{25} - 0 = 1$

Step 1.2.1.1.2

Multiply $- 1$ by $0$ .

$\frac{x^{2}}{25} + 0 = 1$

Step 1.2.1.2

Add $\frac{x^{2}}{25}$ and $0$ .

$\frac{x^{2}}{25} = 1$

Step 1.2.2

Multiply both sides of the equation by $25$ .

$25 \frac{x^{2}}{25} = 25 \cdot 1$

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Cancel the common factor of $25$ .

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Step 1.2.3.1.1.1

Cancel the common factor.

$25 \frac{x^{2}}{25} = 25 \cdot 1$

Step 1.2.3.1.1.2

Rewrite the expression.

$x^{2} = 25 \cdot 1$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $25$ by $1$ .

$x^{2} = 25$

Step 1.2.4

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

$x = \pm \sqrt{25}$

Step 1.2.5

Simplify $\pm \sqrt{25}$ .

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Step 1.2.5.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2}}$

Step 1.2.5.2

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

$x = \pm 5$

Step 1.2.6

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

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Step 1.2.6.1

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

$x = 5$

Step 1.2.6.2

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

$x = - 5$

Step 1.2.6.3

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

$x = 5, - 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(5, 0), (- 5, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$\frac{{(0)}^{2}}{25} - y^{2} = 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $\frac{{(0)}^{2}}{25} - y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{25} - y^{2} = 1$

Step 2.2.1.1.2

Divide $0$ by $25$ .

$0 - y^{2} = 1$

Step 2.2.1.2

Subtract $y^{2}$ from $0$ .

$- y^{2} = 1$

Step 2.2.2

Divide each term in $- y^{2} = 1$ by $- 1$ and simplify.

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Step 2.2.2.1

Divide each term in $- y^{2} = 1$ by $- 1$ .

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

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2.2

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

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

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $1$ by $- 1$ .

$y^{2} = - 1$

Step 2.2.3

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

$y = \pm \sqrt{- 1}$

Step 2.2.4

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \pm i$

Step 2.2.5

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

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Step 2.2.5.1

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

$y = i$

Step 2.2.5.2

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

$y = - i$

Step 2.2.5.3

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

$y = i, - i$

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(5, 0), (- 5, 0)$

y-intercept(s): $None$

Step 4