Precalculus Examples

2 sin (x) + 1 = 0

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

Step 1

Subtract

1

$1$ from both sides of the equation.

2 sin (x) = - 1

$2 \sin (x) = - 1$

Step 2

Divide each term in

2 sin (x) = - 1

$2 \sin (x) = - 1$ by

2

$2$ and simplify.

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Step 2.1

Divide each term in

2 sin (x) = - 1

$2 \sin (x) = - 1$ by

2

$2$ .

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of

2

$2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide

$\sin (x)$ by

$1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

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

Step 3

Take the inverse sine of both sides of the equation to extract

$x$ from inside the sine.

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

Step 4

Simplify the right side.

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Step 4.1

The exact value of

$\arcsin (- \frac{1}{2})$ is

$- \frac{π}{6}$ .

$x = - \frac{π}{6}$

Step 5

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.

$x = 2 π + \frac{π}{6} + π$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Subtract

$2 π$ from

$2 π + \frac{π}{6} + π$ .

$x = 2 π + \frac{π}{6} + π - 2 π$

Step 6.2

The resulting angle of

$\frac{7 π}{6}$ is positive, less than

$2 π$ , and coterminal with

$2 π + \frac{π}{6} + π$ .

$x = \frac{7 π}{6}$

Step 7

Find the period of

$\sin (x)$ .

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Step 7.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 7.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 7.3

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

$0$ and

$1$ is

$1$ .

$\frac{2 π}{1}$

Step 7.4

Divide

$2 π$ by

$1$ .

$2 π$

Step 8

Add

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

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Step 8.1

Add

$2 π$ to

$- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + 2 π$

Step 8.2

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{6}{6}$ .

$2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 8.3

Combine fractions.

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Step 8.3.1

Combine

$2 π$ and

$\frac{6}{6}$ .

$\frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 8.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{6}$

Step 8.4

Simplify the numerator.

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Step 8.4.1

Multiply

$6$ by

$2$ .

$\frac{12 π - π}{6}$

Step 8.4.2

Subtract

$π$ from

$12 π$ .

$\frac{11 π}{6}$

Step 8.5

List the new angles.

$x = \frac{11 π}{6}$

Step 9

The period of the

$\sin (x)$ function is

$2 π$ so values will repeat every

$2 π$ radians in both directions.

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer

$n$