Precalculus Examples

$3 + \csc (x) = \frac{5}{3 - 2 \sin (x)}$

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Subtract $\frac{5}{3 - 2 \sin (x)}$ from both sides of the equation.

$\csc (x) - \frac{5}{3 - 2 \sin (x)} = - 3$

Step 1.2

Rewrite $\csc (x)$ in terms of sines and cosines.

$\frac{1}{\sin (x)} - \frac{5}{3 - 2 \sin (x)} = - 3$

Step 1.3

To write $\frac{1}{\sin (x)}$ as a fraction with a common denominator, multiply by $\frac{3 - 2 \sin (x)}{3 - 2 \sin (x)}$ .

$\frac{1}{\sin (x)} \cdot \frac{3 - 2 \sin (x)}{3 - 2 \sin (x)} - \frac{5}{3 - 2 \sin (x)} = - 3$

Step 1.4

To write $- \frac{5}{3 - 2 \sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

$\frac{1}{\sin (x)} \cdot \frac{3 - 2 \sin (x)}{3 - 2 \sin (x)} - \frac{5}{3 - 2 \sin (x)} \cdot \frac{\sin (x)}{\sin (x)} = - 3$

Step 1.5

Write each expression with a common denominator of $\sin (x) (3 - 2 \sin (x))$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

Multiply $\frac{1}{\sin (x)}$ by $\frac{3 - 2 \sin (x)}{3 - 2 \sin (x)}$ .

$\frac{3 - 2 \sin (x)}{\sin (x) (3 - 2 \sin (x))} - \frac{5}{3 - 2 \sin (x)} \cdot \frac{\sin (x)}{\sin (x)} = - 3$

Step 1.5.2

Multiply $\frac{5}{3 - 2 \sin (x)}$ by $\frac{\sin (x)}{\sin (x)}$ .

$\frac{3 - 2 \sin (x)}{\sin (x) (3 - 2 \sin (x))} - \frac{5 \sin (x)}{(3 - 2 \sin (x)) \sin (x)} = - 3$

Step 1.5.3

Reorder the factors of $(3 - 2 \sin (x)) \sin (x)$ .

$\frac{3 - 2 \sin (x)}{\sin (x) (3 - 2 \sin (x))} - \frac{5 \sin (x)}{\sin (x) (3 - 2 \sin (x))} = - 3$

Step 1.6

Combine the numerators over the common denominator.

$\frac{3 - 2 \sin (x) - 5 \sin (x)}{\sin (x) (3 - 2 \sin (x))} = - 3$

Step 1.7

Subtract $5 \sin (x)$ from $- 2 \sin (x)$ .

$\frac{3 - 7 \sin (x)}{\sin (x) (3 - 2 \sin (x))} = - 3$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$\sin (x) (3 - 2 \sin (x)), 1$

Step 2.2

The LCM of one and any expression is the expression.

$\sin (x) (3 - 2 \sin (x))$

Step 3

Multiply each term in $\frac{3 - 7 \sin (x)}{\sin (x) (3 - 2 \sin (x))} = - 3$ by $\sin (x) (3 - 2 \sin (x))$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{3 - 7 \sin (x)}{\sin (x) (3 - 2 \sin (x))} = - 3$ by $\sin (x) (3 - 2 \sin (x))$ .

$\frac{3 - 7 \sin (x)}{\sin (x) (3 - 2 \sin (x))} (\sin (x) (3 - 2 \sin (x))) = - 3 (\sin (x) (3 - 2 \sin (x)))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $\sin (x) (3 - 2 \sin (x))$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 - 7 \sin (x)}{\sin (x) (3 - 2 \sin (x))} (\sin (x) (3 - 2 \sin (x))) = - 3 (\sin (x) (3 - 2 \sin (x)))$

Step 3.2.1.2

Rewrite the expression.

$3 - 7 \sin (x) = - 3 (\sin (x) (3 - 2 \sin (x)))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify by multiplying through.

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Step 3.3.1.1

Apply the distributive property.

$3 - 7 \sin (x) = - 3 (\sin (x) \cdot 3 + \sin (x) (- 2 \sin (x)))$

Step 3.3.1.2

Reorder.

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Step 3.3.1.2.1

Move $3$ to the left of $\sin (x)$ .

$3 - 7 \sin (x) = - 3 (3 \cdot \sin (x) + \sin (x) (- 2 \sin (x)))$

Step 3.3.1.2.2

Rewrite using the commutative property of multiplication.

$3 - 7 \sin (x) = - 3 (3 \cdot \sin (x) - 2 \sin (x) \sin (x))$

Step 3.3.2

Multiply $\sin (x)$ by $\sin (x)$ by adding the exponents.

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Step 3.3.2.1

Move $\sin (x)$ .

$3 - 7 \sin (x) = - 3 (3 \sin (x) - 2 (\sin (x) \sin (x)))$

Step 3.3.2.2

Multiply $\sin (x)$ by $\sin (x)$ .

$3 - 7 \sin (x) = - 3 (3 \sin (x) - 2 \sin^{2} (x))$

Step 3.3.3

Simplify by multiplying through.

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Step 3.3.3.1

Apply the distributive property.

$3 - 7 \sin (x) = - 3 (3 \sin (x)) - 3 (- 2 \sin^{2} (x))$

Step 3.3.3.2

Multiply.

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Step 3.3.3.2.1

Multiply $3$ by $- 3$ .

$3 - 7 \sin (x) = - 9 \sin (x) - 3 (- 2 \sin^{2} (x))$

Step 3.3.3.2.2

Multiply $- 2$ by $- 3$ .

$3 - 7 \sin (x) = - 9 \sin (x) + 6 \sin^{2} (x)$

Step 4

Solve the equation.

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Step 4.1

Since $\sin (x)$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 9 \sin (x) + 6 \sin^{2} (x) = 3 - 7 \sin (x)$

Step 4.2

Move all terms containing $\sin (x)$ to the left side of the equation.

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Step 4.2.1

Add $7 \sin (x)$ to both sides of the equation.

$- 9 \sin (x) + 6 \sin^{2} (x) + 7 \sin (x) = 3$

Step 4.2.2

Add $- 9 \sin (x)$ and $7 \sin (x)$ .

$6 \sin^{2} (x) - 2 \sin (x) = 3$

Step 4.3

Subtract $3$ from both sides of the equation.

$6 \sin^{2} (x) - 2 \sin (x) - 3 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

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

Step 4.5

Substitute the values $a = 6$ , $b = - 2$ , and $c = - 3$ into the quadratic formula and solve for $\sin (x)$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (6 \cdot - 3)}}{2 \cdot 6}$

Step 4.6

Simplify.

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

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

$\sin (x) = \frac{2 \pm \sqrt{4 - 4 \cdot 6 \cdot - 3}}{2 \cdot 6}$

Step 4.6.1.2

Multiply $- 4 \cdot 6 \cdot - 3$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $6$ .

$\sin (x) = \frac{2 \pm \sqrt{4 - 24 \cdot - 3}}{2 \cdot 6}$

Step 4.6.1.2.2

Multiply $- 24$ by $- 3$ .

$\sin (x) = \frac{2 \pm \sqrt{4 + 72}}{2 \cdot 6}$

Step 4.6.1.3

Add $4$ and $72$ .

$\sin (x) = \frac{2 \pm \sqrt{76}}{2 \cdot 6}$

Step 4.6.1.4

Rewrite $76$ as $2^{2} \cdot 19$ .

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Step 4.6.1.4.1

Factor $4$ out of $76$ .

$\sin (x) = \frac{2 \pm \sqrt{4 (19)}}{2 \cdot 6}$

Step 4.6.1.4.2

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

$\sin (x) = \frac{2 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 6}$

Step 4.6.1.5

Pull terms out from under the radical.

$\sin (x) = \frac{2 \pm 2 \sqrt{19}}{2 \cdot 6}$

Step 4.6.2

Multiply $2$ by $6$ .

$\sin (x) = \frac{2 \pm 2 \sqrt{19}}{12}$

Step 4.6.3

Simplify $\frac{2 \pm 2 \sqrt{19}}{12}$ .

$\sin (x) = \frac{1 \pm \sqrt{19}}{6}$

Step 4.7

The final answer is the combination of both solutions.

$\sin (x) = \frac{1 + \sqrt{19}}{6}, \frac{1 - \sqrt{19}}{6}$

Step 5

Set up each of the solutions to solve for $x$ .

$\sin (x) = \frac{1 + \sqrt{19}}{6}$

$\sin (x) = \frac{1 - \sqrt{19}}{6}$

Step 6

Solve for $x$ in $\sin (x) = \frac{1 + \sqrt{19}}{6}$ .

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Step 6.1

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (\frac{1 + \sqrt{19}}{6})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Evaluate $\arcsin (\frac{1 + \sqrt{19}}{6})$ .

$x = 1.10430054$

Step 6.3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = (3.14159265) - 1.10430054$

Step 6.4

Solve for $x$ .

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Step 6.4.1

Remove parentheses.

$x = 3.14159265 - 1.10430054$

Step 6.4.2

Remove parentheses.

$x = (3.14159265) - 1.10430054$

Step 6.4.3

Subtract $1.10430054$ from $3.14159265$ .

$x = 2.0372921$

Step 6.5

Find the period of $\sin (x)$ .

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Step 6.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 6.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 6.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 6.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.6

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 1.10430054 + 2 π n, 2.0372921 + 2 π n$ , for any integer $n$

Step 7

Solve for $x$ in $\sin (x) = \frac{1 - \sqrt{19}}{6}$ .

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Step 7.1

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (\frac{1 - \sqrt{19}}{6})$

Step 7.2

Simplify the right side.

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Step 7.2.1

Evaluate $\arcsin (\frac{1 - \sqrt{19}}{6})$ .

$x = - 0.59416431$

Step 7.3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = (3.14159265) + 0.59416431$

Step 7.4

Solve for $x$ .

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Step 7.4.1

Remove parentheses.

$x = 3.14159265 + 0.59416431$

Step 7.4.2

Remove parentheses.

$x = (3.14159265) + 0.59416431$

Step 7.4.3

Add $3.14159265$ and $0.59416431$ .

$x = 3.73575697$

Step 7.5

Find the period of $\sin (x)$ .

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Step 7.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 7.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 7.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.6

Add $2 π$ to every negative angle to get positive angles.

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Step 7.6.1

Add $2 π$ to $- 0.59416431$ to find the positive angle.

$- 0.59416431 + 2 π$

Step 7.6.2

Subtract $0.59416431$ from $2 π$ .

$5.68902098$

Step 7.6.3

List the new angles.

$x = 5.68902098$

Step 7.7

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 3.73575697 + 2 π n, 5.68902098 + 2 π n$ , for any integer $n$

Step 8

List all of the solutions.

$x = 1.10430054 + 2 π n, 2.0372921 + 2 π n, 3.73575697 + 2 π n, 5.68902098 + 2 π n$ , for any integer $n$