Trigonometry Examples

$\frac{y^{2}}{4} - \frac{x^{2}}{16} = 1$

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 $x$ .

$\frac{{(0)}^{2}}{4} - \frac{x^{2}}{16} = 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $\frac{{(0)}^{2}}{4} - \frac{x^{2}}{16}$ .

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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 $0$ .

$\frac{0}{4} - \frac{x^{2}}{16} = 1$

Step 1.2.1.1.2

Divide $0$ by $4$ .

$0 - \frac{x^{2}}{16} = 1$

Step 1.2.1.2

Subtract $\frac{x^{2}}{16}$ from $0$ .

$- \frac{x^{2}}{16} = 1$

Step 1.2.2

Multiply both sides of the equation by $- 16$ .

$- 16 (- \frac{x^{2}}{16}) = - 16 \cdot 1$

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

Simplify $- 16 (- \frac{x^{2}}{16})$ .

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Step 1.2.3.1.1.1

Cancel the common factor of $16$ .

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Step 1.2.3.1.1.1.1

Move the leading negative in $- \frac{x^{2}}{16}$ into the numerator.

$- 16 \frac{- x^{2}}{16} = - 16 \cdot 1$

Step 1.2.3.1.1.1.2

Factor $16$ out of $- 16$ .

$16 (- 1) \frac{- x^{2}}{16} = - 16 \cdot 1$

Step 1.2.3.1.1.1.3

Cancel the common factor.

$16 \cdot - 1 \frac{- x^{2}}{16} = - 16 \cdot 1$

Step 1.2.3.1.1.1.4

Rewrite the expression.

$- - x^{2} = - 16 \cdot 1$

Step 1.2.3.1.1.2

Multiply.

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Step 1.2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} = - 16 \cdot 1$

Step 1.2.3.1.1.2.2

Multiply $x^{2}$ by $1$ .

$x^{2} = - 16 \cdot 1$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $- 16$ by $1$ .

$x^{2} = - 16$

Step 1.2.4

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

$x = \pm \sqrt{- 16}$

Step 1.2.5

Simplify $\pm \sqrt{- 16}$ .

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Step 1.2.5.1

Rewrite $- 16$ as $- 1 (16)$ .

$x = \pm \sqrt{- 1 (16)}$

Step 1.2.5.2

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{16}$

Step 1.2.5.3

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

$x = \pm i \cdot \sqrt{16}$

Step 1.2.5.4

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

$x = \pm i \cdot \sqrt{4^{2}}$

Step 1.2.5.5

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

$x = \pm i \cdot 4$

Step 1.2.5.6

Move $4$ to the left of $i$ .

$x = \pm 4 i$

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.

$x = 4 i$

Step 1.2.6.2

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

$x = - 4 i$

Step 1.2.6.3

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

$x = 4 i, - 4 i$

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

$\frac{y^{2}}{4} - \frac{{(0)}^{2}}{16} = 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $\frac{y^{2}}{4} - \frac{{(0)}^{2}}{16}$ .

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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 $0$ .

$\frac{y^{2}}{4} - \frac{0}{16} = 1$

Step 2.2.1.1.2

Divide $0$ by $16$ .

$\frac{y^{2}}{4} - 0 = 1$

Step 2.2.1.1.3

Multiply $- 1$ by $0$ .

$\frac{y^{2}}{4} + 0 = 1$

Step 2.2.1.2

Add $\frac{y^{2}}{4}$ and $0$ .

$\frac{y^{2}}{4} = 1$

Step 2.2.2

Multiply both sides of the equation by $4$ .

$4 \frac{y^{2}}{4} = 4 \cdot 1$

Step 2.2.3

Simplify both sides of the equation.

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Step 2.2.3.1

Simplify the left side.

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Step 2.2.3.1.1

Cancel the common factor of $4$ .

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Step 2.2.3.1.1.1

Cancel the common factor.

$4 \frac{y^{2}}{4} = 4 \cdot 1$

Step 2.2.3.1.1.2

Rewrite the expression.

$y^{2} = 4 \cdot 1$

Step 2.2.3.2

Simplify the right side.

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Step 2.2.3.2.1

Multiply $4$ by $1$ .

$y^{2} = 4$

Step 2.2.4

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

$y = \pm \sqrt{4}$

Step 2.2.5

Simplify $\pm \sqrt{4}$ .

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Step 2.2.5.1

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

$y = \pm \sqrt{2^{2}}$

Step 2.2.5.2

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

$y = \pm 2$

Step 2.2.6

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

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Step 2.2.6.1

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

$y = 2$

Step 2.2.6.2

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

$y = - 2$

Step 2.2.6.3

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

$y = 2, - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2), (0, - 2)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 2), (0, - 2)$

Step 4