Trigonometry Examples

2 sin (\frac{x}{4}) + \sqrt{3} = 0

$2 \sin (\frac{x}{4}) + \sqrt{3} = 0$

Step 1

Subtract

\sqrt{3}

$\sqrt{3}$ from both sides of the equation.

2 sin (\frac{x}{4}) = - \sqrt{3}

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

Step 2

Divide each term in

2 sin (\frac{x}{4}) = - \sqrt{3}

$2 \sin (\frac{x}{4}) = - \sqrt{3}$ by

2

$2$ and simplify.

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Step 2.1

Divide each term in

2 sin (\frac{x}{4}) = - \sqrt{3}

$2 \sin (\frac{x}{4}) = - \sqrt{3}$ by

2

$2$ .

\frac{2 sin (\frac{x}{4})}{2} = \frac{- \sqrt{3}}{2}

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

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.

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

Step 2.2.1.2

Divide

$\sin (\frac{x}{4})$ by

$1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

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

Step 3

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

$x$ from inside the sine.

$\frac{x}{4} = \arcsin (- \frac{\sqrt{3}}{2})$

Step 4

Simplify the right side.

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Step 4.1

The exact value of

$\arcsin (- \frac{\sqrt{3}}{2})$ is

$- \frac{π}{3}$ .

$\frac{x}{4} = - \frac{π}{3}$

Step 5

Multiply both sides of the equation by

$4$ .

$4 \frac{x}{4} = 4 (- \frac{π}{3})$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of

$4$ .

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Step 6.1.1.1

Cancel the common factor.

$4 \frac{x}{4} = 4 (- \frac{π}{3})$

Step 6.1.1.2

Rewrite the expression.

$x = 4 (- \frac{π}{3})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify

$4 (- \frac{π}{3})$ .

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Step 6.2.1.1

Multiply

$4 (- \frac{π}{3})$ .

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Step 6.2.1.1.1

Multiply

$- 1$ by

$4$ .

$x = - 4 \frac{π}{3}$

Step 6.2.1.1.2

Combine

$- 4$ and

$\frac{π}{3}$ .

$x = \frac{- 4 π}{3}$

Step 6.2.1.2

Move the negative in front of the fraction.

$x = - \frac{4 π}{3}$

Step 7

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.

$\frac{x}{4} = 2 π + \frac{π}{3} + π$

Step 8

Simplify the expression to find the second solution.

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Step 8.1

Subtract

$2 π$ from

$2 π + \frac{π}{3} + π$ .

$\frac{x}{4} = 2 π + \frac{π}{3} + π - 2 π$

Step 8.2

The resulting angle of

$\frac{4 π}{3}$ is positive, less than

$2 π$ , and coterminal with

$2 π + \frac{π}{3} + π$ .

$\frac{x}{4} = \frac{4 π}{3}$

Step 8.3

Solve for

$x$ .

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Step 8.3.1

Multiply both sides of the equation by

$4$ .

$4 \frac{x}{4} = 4 \frac{4 π}{3}$

Step 8.3.2

Simplify both sides of the equation.

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Step 8.3.2.1

Simplify the left side.

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Step 8.3.2.1.1

Cancel the common factor of

$4$ .

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Step 8.3.2.1.1.1

Cancel the common factor.

$4 \frac{x}{4} = 4 \frac{4 π}{3}$

Step 8.3.2.1.1.2

Rewrite the expression.

$x = 4 \frac{4 π}{3}$

Step 8.3.2.2

Simplify the right side.

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Step 8.3.2.2.1

Multiply

$4 \frac{4 π}{3}$ .

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Step 8.3.2.2.1.1

Combine

$4$ and

$\frac{4 π}{3}$ .

$x = \frac{4 (4 π)}{3}$

Step 8.3.2.2.1.2

Multiply

$4$ by

$4$ .

$x = \frac{16 π}{3}$

Step 9

Find the period of

$\sin (\frac{x}{4})$ .

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Step 9.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 9.2

Replace

$b$ with

$\frac{1}{4}$ in the formula for period.

$\frac{2 π}{| \frac{1}{4} |}$

Step 9.3

$\frac{1}{4}$ is approximately

$0.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{4}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 4$

Step 9.5

Multiply

$4$ by

$2$ .

$8 π$

Step 10

Add

$8 π$ to every negative angle to get positive angles.

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Step 10.1

Add

$8 π$ to

$- \frac{4 π}{3}$ to find the positive angle.

$- \frac{4 π}{3} + 8 π$

Step 10.2

To write

$8 π$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$8 π \cdot \frac{3}{3} - \frac{4 π}{3}$

Step 10.3

Combine fractions.

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Step 10.3.1

Combine

$8 π$ and

$\frac{3}{3}$ .

$\frac{8 π \cdot 3}{3} - \frac{4 π}{3}$

Step 10.3.2

Combine the numerators over the common denominator.

$\frac{8 π \cdot 3 - 4 π}{3}$

Step 10.4

Simplify the numerator.

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Step 10.4.1

Multiply

$3$ by

$8$ .

$\frac{24 π - 4 π}{3}$

Step 10.4.2

Subtract

$4 π$ from

$24 π$ .

$\frac{20 π}{3}$

Step 10.5

List the new angles.

$x = \frac{20 π}{3}$

Step 11

The period of the

$\sin (\frac{x}{4})$ function is

$8 π$ so values will repeat every

$8 π$ radians in both directions.

$x = \frac{16 π}{3} + 8 π n, \frac{20 π}{3} + 8 π n$ , for any integer

$n$