Precalculus Examples

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

Step 1

Substitute $u$ for $\sin (x)$ .

${(u)}^{2} + u = \frac{1}{2}$

Step 2

Subtract $\frac{1}{2}$ from both sides of the equation.

$u^{2} + u - \frac{1}{2} = 0$

Step 3

Multiply through by the least common denominator $2$ , then simplify.

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Step 3.1

Apply the distributive property.

$2 u^{2} + 2 u + 2 (- \frac{1}{2}) = 0$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$2 u^{2} + 2 u + 2 (\frac{- 1}{2}) = 0$

Step 3.2.2

Cancel the common factor.

$2 u^{2} + 2 u + 2 (\frac{- 1}{2}) = 0$

Step 3.2.3

Rewrite the expression.

$2 u^{2} + 2 u - 1 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 2$ , $b = 2$ , and $c = - 1$ into the quadratic formula and solve for $u$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (2 \cdot - 1)}}{2 \cdot 2}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$u = \frac{- 2 \pm \sqrt{4 - 8 \cdot - 1}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $- 1$ .

$u = \frac{- 2 \pm \sqrt{4 + 8}}{2 \cdot 2}$

Step 6.1.3

Add $4$ and $8$ .

$u = \frac{- 2 \pm \sqrt{12}}{2 \cdot 2}$

Step 6.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 6.1.4.1

Factor $4$ out of $12$ .

$u = \frac{- 2 \pm \sqrt{4 (3)}}{2 \cdot 2}$

Step 6.1.4.2

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

$u = \frac{- 2 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 2}$

Step 6.1.5

Pull terms out from under the radical.

$u = \frac{- 2 \pm 2 \sqrt{3}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$u = \frac{- 2 \pm 2 \sqrt{3}}{4}$

Step 6.3

Simplify $\frac{- 2 \pm 2 \sqrt{3}}{4}$ .

$u = \frac{- 1 \pm \sqrt{3}}{2}$

Step 7

The final answer is the combination of both solutions.

$u = - \frac{1 - \sqrt{3}}{2}, - \frac{1 + \sqrt{3}}{2}$

Step 8

Substitute $\sin (x)$ for $u$ .

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

Step 9

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

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

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

Step 10

Solve for $x$ in $\sin (x) = - \frac{1 - \sqrt{3}}{2}$ .

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Step 10.1

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

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

Step 10.2

Simplify the right side.

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Step 10.2.1

Evaluate $\arcsin (- \frac{1 - \sqrt{3}}{2})$ .

$x = 0.37473443$

Step 10.3

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.

$x = 2 (3.14159265) - 0.37473443 + 3.14159265$

Step 10.4

Simplify the expression to find the second solution.

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Step 10.4.1

Subtract $2 π$ from $2 (3.14159265) - 0.37473443 + 3.14159265$ .

$x = 2 (3.14159265) - 0.37473443 + 3.14159265 - 2 π$

Step 10.4.2

The resulting angle of $2.76685822$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) - 0.37473443 + 3.14159265$ .

$x = 2.76685822$

Step 10.5

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

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Step 10.5.1

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

$\frac{2 π}{| b |}$

Step 10.5.2

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

$\frac{2 π}{| 1 |}$

Step 10.5.3

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

$\frac{2 π}{1}$

Step 10.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 10.6

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

$x = 0.37473443 + 2 π n, 2.76685822 + 2 π n$ , for any integer $n$

Step 11

Solve for $x$ in $\sin (x) = - \frac{1 + \sqrt{3}}{2}$ .

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Step 11.1

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

No solution

Step 12

List all of the solutions.

$x = 0.37473443 + 2 π n, 2.76685822 + 2 π n$ , for any integer $n$