Trigonometry Examples

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

Step 1

Divide each term in $4 \sin (\frac{x}{2}) = 2 \sqrt{3}$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 \sin (\frac{x}{2}) = 2 \sqrt{3}$ by $4$ .

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $4$ .

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Divide $\sin (\frac{x}{2})$ by $1$ .

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $2$ and $4$ .

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Step 1.3.1.1

Factor $2$ out of $2 \sqrt{3}$ .

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

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $2$ out of $4$ .

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

Step 1.3.1.2.2

Cancel the common factor.

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

Step 1.3.1.2.3

Rewrite the expression.

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

Step 2

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

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

Step 3

Simplify the right side.

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Step 3.1

The exact value of $\arcsin (\frac{\sqrt{3}}{2})$ is $\frac{π}{3}$ .

$\frac{x}{2} = \frac{π}{3}$

Step 4

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{π}{3}$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{π}{3}$

Step 5.1.1.2

Rewrite the expression.

$x = 2 \frac{π}{3}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Combine $2$ and $\frac{π}{3}$ .

$x = \frac{2 π}{3}$

Step 6

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$\frac{x}{2} = π - \frac{π}{3}$

Step 7

Solve for $x$ .

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Step 7.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π - \frac{π}{3})$

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Cancel the common factor of $2$ .

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Step 7.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π - \frac{π}{3})$

Step 7.2.1.1.2

Rewrite the expression.

$x = 2 (π - \frac{π}{3})$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $2 (π - \frac{π}{3})$ .

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Step 7.2.2.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 2 (π \cdot \frac{3}{3} - \frac{π}{3})$

Step 7.2.2.1.2

Combine fractions.

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Step 7.2.2.1.2.1

Combine $π$ and $\frac{3}{3}$ .

$x = 2 (\frac{π \cdot 3}{3} - \frac{π}{3})$

Step 7.2.2.1.2.2

Combine the numerators over the common denominator.

$x = 2 \frac{π \cdot 3 - π}{3}$

Step 7.2.2.1.3

Simplify the numerator.

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Step 7.2.2.1.3.1

Move $3$ to the left of $π$ .

$x = 2 \frac{3 \cdot π - π}{3}$

Step 7.2.2.1.3.2

Subtract $π$ from $3 π$ .

$x = 2 \frac{2 π}{3}$

Step 7.2.2.1.4

Multiply $2 \frac{2 π}{3}$ .

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Step 7.2.2.1.4.1

Combine $2$ and $\frac{2 π}{3}$ .

$x = \frac{2 (2 π)}{3}$

Step 7.2.2.1.4.2

Multiply $2$ by $2$ .

$x = \frac{4 π}{3}$

Step 8

Find the period of $\sin (\frac{x}{2})$ .

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 8.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 8.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 8.5

Multiply $2$ by $2$ .

$4 π$

Step 9

The period of the $\sin (\frac{x}{2})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = \frac{2 π}{3} + 4 π n, \frac{4 π}{3} + 4 π n$ , for any integer $n$