Trigonometry Examples

$\frac{\frac{10 r^{2} - 94 r + 36}{5 r^{2} + 23 r - 10}}{\frac{45 r^{2} - 23 r + 4}{9 r^{2} + 43 r - 10}}$

Step 1

Set the denominator in $\frac{10 r^{2} - 94 r + 36}{5 r^{2} + 23 r - 10}$ equal to $0$ to find where the expression is undefined.

$5 r^{2} + 23 r - 10 = 0$

Step 2

Solve for $r$ .

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Step 2.1

Factor by grouping.

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Step 2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 10 = - 50$ and whose sum is $b = 23$ .

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Step 2.1.1.1

Factor $23$ out of $23 r$ .

$5 r^{2} + 23 (r) - 10 = 0$

Step 2.1.1.2

Rewrite $23$ as $- 2$ plus $25$

$5 r^{2} + (- 2 + 25) r - 10 = 0$

Step 2.1.1.3

Apply the distributive property.

$5 r^{2} - 2 r + 25 r - 10 = 0$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$(5 r^{2} - 2 r) + 25 r - 10 = 0$

Step 2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$r (5 r - 2) + 5 (5 r - 2) = 0$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 r - 2$ .

$(5 r - 2) (r + 5) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$5 r - 2 = 0$

$r + 5 = 0$

Step 2.3

Set $5 r - 2$ equal to $0$ and solve for $r$ .

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Step 2.3.1

Set $5 r - 2$ equal to $0$ .

$5 r - 2 = 0$

Step 2.3.2

Solve $5 r - 2 = 0$ for $r$ .

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Step 2.3.2.1

Add $2$ to both sides of the equation.

$5 r = 2$

Step 2.3.2.2

Divide each term in $5 r = 2$ by $5$ and simplify.

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Step 2.3.2.2.1

Divide each term in $5 r = 2$ by $5$ .

$\frac{5 r}{5} = \frac{2}{5}$

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.3.2.2.2.1.1

Cancel the common factor.

$\frac{5 r}{5} = \frac{2}{5}$

Step 2.3.2.2.2.1.2

Divide $r$ by $1$ .

$r = \frac{2}{5}$

Step 2.4

Set $r + 5$ equal to $0$ and solve for $r$ .

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Step 2.4.1

Set $r + 5$ equal to $0$ .

$r + 5 = 0$

Step 2.4.2

Subtract $5$ from both sides of the equation.

$r = - 5$

Step 2.5

The final solution is all the values that make $(5 r - 2) (r + 5) = 0$ true.

$r = \frac{2}{5}, - 5$

Step 3

Set the denominator in $\frac{45 r^{2} - 23 r + 4}{9 r^{2} + 43 r - 10}$ equal to $0$ to find where the expression is undefined.

$9 r^{2} + 43 r - 10 = 0$

Step 4

Solve for $r$ .

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Step 4.1

Factor by grouping.

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Step 4.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot - 10 = - 90$ and whose sum is $b = 43$ .

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Step 4.1.1.1

Factor $43$ out of $43 r$ .

$9 r^{2} + 43 (r) - 10 = 0$

Step 4.1.1.2

Rewrite $43$ as $- 2$ plus $45$

$9 r^{2} + (- 2 + 45) r - 10 = 0$

Step 4.1.1.3

Apply the distributive property.

$9 r^{2} - 2 r + 45 r - 10 = 0$

Step 4.1.2

Factor out the greatest common factor from each group.

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Step 4.1.2.1

Group the first two terms and the last two terms.

$(9 r^{2} - 2 r) + 45 r - 10 = 0$

Step 4.1.2.2

Factor out the greatest common factor (GCF) from each group.

$r (9 r - 2) + 5 (9 r - 2) = 0$

Step 4.1.3

Factor the polynomial by factoring out the greatest common factor, $9 r - 2$ .

$(9 r - 2) (r + 5) = 0$

Step 4.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$9 r - 2 = 0$

$r + 5 = 0$

Step 4.3

Set $9 r - 2$ equal to $0$ and solve for $r$ .

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Step 4.3.1

Set $9 r - 2$ equal to $0$ .

$9 r - 2 = 0$

Step 4.3.2

Solve $9 r - 2 = 0$ for $r$ .

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Step 4.3.2.1

Add $2$ to both sides of the equation.

$9 r = 2$

Step 4.3.2.2

Divide each term in $9 r = 2$ by $9$ and simplify.

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Step 4.3.2.2.1

Divide each term in $9 r = 2$ by $9$ .

$\frac{9 r}{9} = \frac{2}{9}$

Step 4.3.2.2.2

Simplify the left side.

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Step 4.3.2.2.2.1

Cancel the common factor of $9$ .

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Step 4.3.2.2.2.1.1

Cancel the common factor.

$\frac{9 r}{9} = \frac{2}{9}$

Step 4.3.2.2.2.1.2

Divide $r$ by $1$ .

$r = \frac{2}{9}$

Step 4.4

Set $r + 5$ equal to $0$ and solve for $r$ .

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Step 4.4.1

Set $r + 5$ equal to $0$ .

$r + 5 = 0$

Step 4.4.2

Subtract $5$ from both sides of the equation.

$r = - 5$

Step 4.5

The final solution is all the values that make $(9 r - 2) (r + 5) = 0$ true.

$r = \frac{2}{9}, - 5$

Step 5

Set the denominator in $\frac{\frac{10 r^{2} - 94 r + 36}{5 r^{2} + 23 r - 10}}{\frac{45 r^{2} - 23 r + 4}{9 r^{2} + 43 r - 10}}$ equal to $0$ to find where the expression is undefined.

$\frac{45 r^{2} - 23 r + 4}{9 r^{2} + 43 r - 10} = 0$

Step 6

Solve for $r$ .

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Step 6.1

Set the numerator equal to zero.

$45 r^{2} - 23 r + 4 = 0$

Step 6.2

Solve the equation for $r$ .

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Step 6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.2.2

Substitute the values $a = 45$ , $b = - 23$ , and $c = 4$ into the quadratic formula and solve for $r$ .

$\frac{23 \pm \sqrt{{(- 23)}^{2} - 4 \cdot (45 \cdot 4)}}{2 \cdot 45}$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

Raise $- 23$ to the power of $2$ .

$r = \frac{23 \pm \sqrt{529 - 4 \cdot 45 \cdot 4}}{2 \cdot 45}$

Step 6.2.3.1.2

Multiply $- 4 \cdot 45 \cdot 4$ .

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Step 6.2.3.1.2.1

Multiply $- 4$ by $45$ .

$r = \frac{23 \pm \sqrt{529 - 180 \cdot 4}}{2 \cdot 45}$

Step 6.2.3.1.2.2

Multiply $- 180$ by $4$ .

$r = \frac{23 \pm \sqrt{529 - 720}}{2 \cdot 45}$

Step 6.2.3.1.3

Subtract $720$ from $529$ .

$r = \frac{23 \pm \sqrt{- 191}}{2 \cdot 45}$

Step 6.2.3.1.4

Rewrite $- 191$ as $- 1 (191)$ .

$r = \frac{23 \pm \sqrt{- 1 \cdot 191}}{2 \cdot 45}$

Step 6.2.3.1.5

Rewrite $\sqrt{- 1 (191)}$ as $\sqrt{- 1} \cdot \sqrt{191}$ .

$r = \frac{23 \pm \sqrt{- 1} \cdot \sqrt{191}}{2 \cdot 45}$

Step 6.2.3.1.6

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

$r = \frac{23 \pm i \sqrt{191}}{2 \cdot 45}$

Step 6.2.3.2

Multiply $2$ by $45$ .

$r = \frac{23 \pm i \sqrt{191}}{90}$

Step 6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.2.4.1

Simplify the numerator.

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Step 6.2.4.1.1

Raise $- 23$ to the power of $2$ .

$r = \frac{23 \pm \sqrt{529 - 4 \cdot 45 \cdot 4}}{2 \cdot 45}$

Step 6.2.4.1.2

Multiply $- 4 \cdot 45 \cdot 4$ .

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Step 6.2.4.1.2.1

Multiply $- 4$ by $45$ .

$r = \frac{23 \pm \sqrt{529 - 180 \cdot 4}}{2 \cdot 45}$

Step 6.2.4.1.2.2

Multiply $- 180$ by $4$ .

$r = \frac{23 \pm \sqrt{529 - 720}}{2 \cdot 45}$

Step 6.2.4.1.3

Subtract $720$ from $529$ .

$r = \frac{23 \pm \sqrt{- 191}}{2 \cdot 45}$

Step 6.2.4.1.4

Rewrite $- 191$ as $- 1 (191)$ .

$r = \frac{23 \pm \sqrt{- 1 \cdot 191}}{2 \cdot 45}$

Step 6.2.4.1.5

Rewrite $\sqrt{- 1 (191)}$ as $\sqrt{- 1} \cdot \sqrt{191}$ .

$r = \frac{23 \pm \sqrt{- 1} \cdot \sqrt{191}}{2 \cdot 45}$

Step 6.2.4.1.6

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

$r = \frac{23 \pm i \sqrt{191}}{2 \cdot 45}$

Step 6.2.4.2

Multiply $2$ by $45$ .

$r = \frac{23 \pm i \sqrt{191}}{90}$

Step 6.2.4.3

Change the $\pm$ to $+$ .

$r = \frac{23 + i \sqrt{191}}{90}$

Step 6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.2.5.1

Simplify the numerator.

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Step 6.2.5.1.1

Raise $- 23$ to the power of $2$ .

$r = \frac{23 \pm \sqrt{529 - 4 \cdot 45 \cdot 4}}{2 \cdot 45}$

Step 6.2.5.1.2

Multiply $- 4 \cdot 45 \cdot 4$ .

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Step 6.2.5.1.2.1

Multiply $- 4$ by $45$ .

$r = \frac{23 \pm \sqrt{529 - 180 \cdot 4}}{2 \cdot 45}$

Step 6.2.5.1.2.2

Multiply $- 180$ by $4$ .

$r = \frac{23 \pm \sqrt{529 - 720}}{2 \cdot 45}$

Step 6.2.5.1.3

Subtract $720$ from $529$ .

$r = \frac{23 \pm \sqrt{- 191}}{2 \cdot 45}$

Step 6.2.5.1.4

Rewrite $- 191$ as $- 1 (191)$ .

$r = \frac{23 \pm \sqrt{- 1 \cdot 191}}{2 \cdot 45}$

Step 6.2.5.1.5

Rewrite $\sqrt{- 1 (191)}$ as $\sqrt{- 1} \cdot \sqrt{191}$ .

$r = \frac{23 \pm \sqrt{- 1} \cdot \sqrt{191}}{2 \cdot 45}$

Step 6.2.5.1.6

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

$r = \frac{23 \pm i \sqrt{191}}{2 \cdot 45}$

Step 6.2.5.2

Multiply $2$ by $45$ .

$r = \frac{23 \pm i \sqrt{191}}{90}$

Step 6.2.5.3

Change the $\pm$ to $-$ .

$r = \frac{23 - i \sqrt{191}}{90}$

Step 6.2.6

The final answer is the combination of both solutions.

$r = \frac{23 + i \sqrt{191}}{90}, \frac{23 - i \sqrt{191}}{90}$

Step 7

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$r = - 5, r = \frac{2}{9}, r = \frac{2}{5}$

Step 8