Precalculus Examples

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

Step 1

Add $1$ to both sides of the equation.

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

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.

$1, \sin (x), 1$

Step 2.2

The LCM of one and any expression is the expression.

$\sin (x)$

Step 3

Multiply each term in $2 \sin (x) = \frac{3}{\sin (x)} + 1$ by $\sin (x)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $2 \sin (x) = \frac{3}{\sin (x)} + 1$ by $\sin (x)$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Move $\sin (x)$ .

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

Step 3.2.1.2

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

$2 \sin^{2} (x) = \frac{3}{\sin (x)} \sin (x) + 1 \sin (x)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

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

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Step 3.3.1.1.1

Cancel the common factor.

$2 \sin^{2} (x) = \frac{3}{\sin (x)} \sin (x) + 1 \sin (x)$

Step 3.3.1.1.2

Rewrite the expression.

$2 \sin^{2} (x) = 3 + 1 \sin (x)$

Step 3.3.1.2

Multiply $\sin (x)$ by $1$ .

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

Step 4

Solve the equation.

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Step 4.1

Subtract $\sin (x)$ from both sides of the equation.

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

Step 4.2

Subtract $3$ from both sides of the equation.

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

Step 4.3

Factor by grouping.

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Step 4.3.1

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 1$ .

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Step 4.3.1.1

Factor $- 1$ out of $- \sin (x)$ .

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

Step 4.3.1.2

Rewrite $- 1$ as $2$ plus $- 3$

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

Step 4.3.1.3

Apply the distributive property.

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

Step 4.3.2

Factor out the greatest common factor from each group.

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Step 4.3.2.1

Group the first two terms and the last two terms.

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

Step 4.3.2.2

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

$2 \sin (x) (\sin (x) + 1) - 3 (\sin (x) + 1) = 0$

Step 4.3.3

Factor the polynomial by factoring out the greatest common factor, $\sin (x) + 1$ .

$(\sin (x) + 1) (2 \sin (x) - 3) = 0$

Step 4.4

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

$\sin (x) + 1 = 0$

$2 \sin (x) - 3 = 0$

Step 4.5

Set $\sin (x) + 1$ equal to $0$ and solve for $\sin (x)$ .

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Step 4.5.1

Set $\sin (x) + 1$ equal to $0$ .

$\sin (x) + 1 = 0$

Step 4.5.2

Subtract $1$ from both sides of the equation.

$\sin (x) = - 1$

Step 4.6

Set $2 \sin (x) - 3$ equal to $0$ and solve for $\sin (x)$ .

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Step 4.6.1

Set $2 \sin (x) - 3$ equal to $0$ .

$2 \sin (x) - 3 = 0$

Step 4.6.2

Solve $2 \sin (x) - 3 = 0$ for $\sin (x)$ .

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Step 4.6.2.1

Add $3$ to both sides of the equation.

$2 \sin (x) = 3$

Step 4.6.2.2

Divide each term in $2 \sin (x) = 3$ by $2$ and simplify.

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Step 4.6.2.2.1

Divide each term in $2 \sin (x) = 3$ by $2$ .

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

Step 4.6.2.2.2

Simplify the left side.

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Step 4.6.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.6.2.2.2.1.1

Cancel the common factor.

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

Step 4.6.2.2.2.1.2

Divide $\sin (x)$ by $1$ .

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

Step 4.7

The final solution is all the values that make $(\sin (x) + 1) (2 \sin (x) - 3) = 0$ true.

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

Step 5

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

$\sin (x) = - 1$

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

Step 6

Solve for $x$ in $\sin (x) = - 1$ .

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Step 6.1

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

$x = \arcsin (- 1)$

Step 6.2

Simplify the right side.

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Step 6.2.1

The exact value of $\arcsin (- 1)$ is $- \frac{π}{2}$ .

$x = - \frac{π}{2}$

Step 6.3

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{2} + π$

Step 6.4

Simplify the expression to find the second solution.

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Step 6.4.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 6.4.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$x = \frac{3 π}{2}$

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

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

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Step 6.6.1

Add $2 π$ to $- \frac{π}{2}$ to find the positive angle.

$- \frac{π}{2} + 2 π$

Step 6.6.2

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

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 6.6.3

Combine fractions.

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Step 6.6.3.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 6.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Step 6.6.4

Simplify the numerator.

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Step 6.6.4.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Step 6.6.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

Step 6.6.5

List the new angles.

$x = \frac{3 π}{2}$

Step 6.7

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

$x = \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 7

Solve for $x$ in $\sin (x) = \frac{3}{2}$ .

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Step 7.1

The range of sine is $- 1 \leq y \leq 1$ . Since $\frac{3}{2}$ does not fall in this range, there is no solution.

No solution

Step 8

List all of the solutions.

$x = \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 9

Consolidate the answers.

$x = \frac{3 π}{2} + 2 π n$ , for any integer $n$