Precalculus Examples

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

Step 1

Replace the $- 2 \cos^{2} (x)$ with $- 2 (1 - \sin^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$\sin (x) - 2 (1 - \sin^{2} (x)) = 0$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$\sin (x) - 2 \cdot 1 - 2 (- \sin^{2} (x)) = 0$

Step 2.2

Multiply $- 2$ by $1$ .

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

Step 2.3

Multiply $- 1$ by $- 2$ .

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

Step 3

Reorder the polynomial.

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

Step 4

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

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

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

One to any power is one.

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

Step 7.1.2

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

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

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

Step 7.1.2.2

Multiply $- 8$ by $- 2$ .

$u = \frac{- 1 \pm \sqrt{1 + 16}}{2 \cdot 2}$

Step 7.1.3

Add $1$ and $16$ .

$u = \frac{- 1 \pm \sqrt{17}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$u = \frac{- 1 \pm \sqrt{17}}{4}$

Step 8

The final answer is the combination of both solutions.

$u = - \frac{1 - \sqrt{17}}{4}, - \frac{1 + \sqrt{17}}{4}$

Step 9

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

$\sin (x) = - \frac{1 - \sqrt{17}}{4}, - \frac{1 + \sqrt{17}}{4}$

Step 10

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

$\sin (x) = - \frac{1 - \sqrt{17}}{4}$

$\sin (x) = - \frac{1 + \sqrt{17}}{4}$

Step 11

Solve for $x$ in $\sin (x) = - \frac{1 - \sqrt{17}}{4}$ .

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Step 11.1

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

$x = \arcsin (- \frac{1 - \sqrt{17}}{4})$

Step 11.2

Simplify the right side.

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Step 11.2.1

Evaluate $\arcsin (- \frac{1 - \sqrt{17}}{4})$ .

$x = 0.89590748$

Step 11.3

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.

$x = 2 (3.14159265) - 0.89590748 + 3.14159265$

Step 11.4

Simplify the expression to find the second solution.

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Step 11.4.1

Subtract $2 π$ from $2 (3.14159265) - 0.89590748 + 3.14159265$ .

$x = 2 (3.14159265) - 0.89590748 + 3.14159265 - 2 π$

Step 11.4.2

The resulting angle of $2.24568517$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) - 0.89590748 + 3.14159265$ .

$x = 2.24568517$

Step 11.5

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

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Step 11.5.1

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

$\frac{2 π}{| b |}$

Step 11.5.2

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

$\frac{2 π}{| 1 |}$

Step 11.5.3

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

$\frac{2 π}{1}$

Step 11.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 11.6

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

$x = 0.89590748 + 2 π n, 2.24568517 + 2 π n$ , for any integer $n$

Step 12

Solve for $x$ in $\sin (x) = - \frac{1 + \sqrt{17}}{4}$ .

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Step 12.1

The range of sine is $- 1 \leq y \leq 1$ . Since $- \frac{1 + \sqrt{17}}{4}$ does not fall in this range, there is no solution.

No solution

Step 13

List all of the solutions.

$x = 0.89590748 + 2 π n, 2.24568517 + 2 π n$ , for any integer $n$