Trigonometry Examples

$5 x^{2} + 5 y^{2} - 12 y = 2$

Step 1

Divide both sides of the equation by $5$ .

$x^{2} + y^{2} - \frac{12 y}{5} = \frac{2}{5}$

Step 2

Complete the square for $y^{2} - \frac{12 y}{5}$ .

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Step 2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = - \frac{12}{5}$

$c = 0$

Step 2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- \frac{12}{5}}{2 \cdot 1}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $- 1$ and $1$ .

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Step 2.3.2.1.1

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

$d = \frac{- 1 (1) \frac{12}{5}}{2 \cdot 1}$

Step 2.3.2.1.2

Cancel the common factor.

$d = \frac{- 1 \cdot 1 \frac{12}{5}}{2 \cdot 1}$

Step 2.3.2.1.3

Rewrite the expression.

$d = \frac{- \frac{12}{5}}{2}$

Step 2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{12}{5} \cdot \frac{1}{2}$

Step 2.3.2.3

Cancel the common factor of $2$ .

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Step 2.3.2.3.1

Move the leading negative in $- \frac{12}{5}$ into the numerator.

$d = \frac{- 12}{5} \cdot \frac{1}{2}$

Step 2.3.2.3.2

Factor $2$ out of $- 12$ .

$d = \frac{2 (- 6)}{5} \cdot \frac{1}{2}$

Step 2.3.2.3.3

Cancel the common factor.

$d = \frac{2 \cdot - 6}{5} \cdot \frac{1}{2}$

Step 2.3.2.3.4

Rewrite the expression.

$d = \frac{- 6}{5}$

Step 2.3.2.4

Move the negative in front of the fraction.

$d = - \frac{6}{5}$

Step 2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- \frac{12}{5})}^{2}}{4 \cdot 1}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Simplify the numerator.

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Step 2.4.2.1.1.1

Apply the product rule to $- \frac{12}{5}$ .

$e = 0 - \frac{{(- 1)}^{2} {(\frac{12}{5})}^{2}}{4 \cdot 1}$

Step 2.4.2.1.1.2

Raise $- 1$ to the power of $2$ .

$e = 0 - \frac{1 {(\frac{12}{5})}^{2}}{4 \cdot 1}$

Step 2.4.2.1.1.3

Apply the product rule to $\frac{12}{5}$ .

$e = 0 - \frac{1 \frac{12^{2}}{5^{2}}}{4 \cdot 1}$

Step 2.4.2.1.1.4

Raise $12$ to the power of $2$ .

$e = 0 - \frac{1 \frac{144}{5^{2}}}{4 \cdot 1}$

Step 2.4.2.1.1.5

Raise $5$ to the power of $2$ .

$e = 0 - \frac{1 (\frac{144}{25})}{4 \cdot 1}$

Step 2.4.2.1.1.6

Multiply $\frac{144}{25}$ by $1$ .

$e = 0 - \frac{\frac{144}{25}}{4 \cdot 1}$

Step 2.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{\frac{144}{25}}{4}$

Step 2.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (\frac{144}{25} \cdot \frac{1}{4})$

Step 2.4.2.1.4

Cancel the common factor of $4$ .

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Step 2.4.2.1.4.1

Factor $4$ out of $144$ .

$e = 0 - (\frac{4 (36)}{25} \cdot \frac{1}{4})$

Step 2.4.2.1.4.2

Cancel the common factor.

$e = 0 - (\frac{4 \cdot 36}{25} \cdot \frac{1}{4})$

Step 2.4.2.1.4.3

Rewrite the expression.

$e = 0 - \frac{36}{25}$

Step 2.4.2.2

Subtract $\frac{36}{25}$ from $0$ .

$e = - \frac{36}{25}$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - \frac{6}{5})}^{2} - \frac{36}{25}$ .

${(y - \frac{6}{5})}^{2} - \frac{36}{25}$

Step 3

Substitute ${(y - \frac{6}{5})}^{2} - \frac{36}{25}$ for $y^{2} - \frac{12 y}{5}$ in the equation $x^{2} + y^{2} - \frac{12 y}{5} = \frac{2}{5}$ .

$x^{2} + {(y - \frac{6}{5})}^{2} - \frac{36}{25} = \frac{2}{5}$

Step 4

Move $- \frac{36}{25}$ to the right side of the equation by adding $\frac{36}{25}$ to both sides.

$x^{2} + {(y - \frac{6}{5})}^{2} = \frac{2}{5} + \frac{36}{25}$

Step 5

Simplify $\frac{2}{5} + \frac{36}{25}$ .

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Step 5.1

To write $\frac{2}{5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{2} + {(y - \frac{6}{5})}^{2} = \frac{2}{5} \cdot \frac{5}{5} + \frac{36}{25}$

Step 5.2

Write each expression with a common denominator of $25$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1

Multiply $\frac{2}{5}$ by $\frac{5}{5}$ .

$x^{2} + {(y - \frac{6}{5})}^{2} = \frac{2 \cdot 5}{5 \cdot 5} + \frac{36}{25}$

Step 5.2.2

Multiply $5$ by $5$ .

$x^{2} + {(y - \frac{6}{5})}^{2} = \frac{2 \cdot 5}{25} + \frac{36}{25}$

Step 5.3

Combine the numerators over the common denominator.

$x^{2} + {(y - \frac{6}{5})}^{2} = \frac{2 \cdot 5 + 36}{25}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $2$ by $5$ .

$x^{2} + {(y - \frac{6}{5})}^{2} = \frac{10 + 36}{25}$

Step 5.4.2

Add $10$ and $36$ .

$x^{2} + {(y - \frac{6}{5})}^{2} = \frac{46}{25}$

Step 6

This is the form of a circle. Use this form to determine the center and radius of the circle.

${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$

Step 7

Match the values in this circle to those of the standard form. The variable $r$ represents the radius of the circle, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from origin.

$r = \frac{\sqrt{46}}{5}$

$h = 0$

$k = \frac{6}{5}$

Step 8

The center of the circle is found at $(h, k)$ .

Center: $(0, \frac{6}{5})$

Step 9

These values represent the important values for graphing and analyzing a circle.

Center: $(0, \frac{6}{5})$

Radius: $\frac{\sqrt{46}}{5}$

Step 10