Trigonometry Examples

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

Step 1

Move all terms containing $\sin (\frac{x}{2})$ to the left side of the equation.

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Step 1.1

Add $\sin (\frac{x}{2})$ to both sides of the equation.

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

Step 1.2

Add $\sin (\frac{x}{2})$ and $\sin (\frac{x}{2})$ .

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

Step 2

Divide each term in $2 \sin (\frac{x}{2}) = 1$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 \sin (\frac{x}{2}) = 1$ by $2$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

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

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

Step 3

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

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

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$\frac{x}{2} = \frac{π}{6}$

Step 5

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{π}{6}$

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 $2$ .

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Step 6.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{π}{6}$

Step 6.1.1.2

Rewrite the expression.

$x = 2 \frac{π}{6}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Factor $2$ out of $6$ .

$x = 2 \frac{π}{2 (3)}$

Step 6.2.1.2

Cancel the common factor.

$x = 2 \frac{π}{2 \cdot 3}$

Step 6.2.1.3

Rewrite the expression.

$x = \frac{π}{3}$

Step 7

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.

$\frac{x}{2} = π - \frac{π}{6}$

Step 8

Solve for $x$ .

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Step 8.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π - \frac{π}{6})$

Step 8.2

Simplify both sides of the equation.

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Step 8.2.1

Simplify the left side.

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Step 8.2.1.1

Cancel the common factor of $2$ .

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Step 8.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π - \frac{π}{6})$

Step 8.2.1.1.2

Rewrite the expression.

$x = 2 (π - \frac{π}{6})$

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

Simplify $2 (π - \frac{π}{6})$ .

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Step 8.2.2.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = 2 (π \cdot \frac{6}{6} - \frac{π}{6})$

Step 8.2.2.1.2

Simplify terms.

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Step 8.2.2.1.2.1

Combine $π$ and $\frac{6}{6}$ .

$x = 2 (\frac{π \cdot 6}{6} - \frac{π}{6})$

Step 8.2.2.1.2.2

Combine the numerators over the common denominator.

$x = 2 \frac{π \cdot 6 - π}{6}$

Step 8.2.2.1.2.3

Cancel the common factor of $2$ .

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Step 8.2.2.1.2.3.1

Factor $2$ out of $6$ .

$x = 2 \frac{π \cdot 6 - π}{2 (3)}$

Step 8.2.2.1.2.3.2

Cancel the common factor.

$x = 2 \frac{π \cdot 6 - π}{2 \cdot 3}$

Step 8.2.2.1.2.3.3

Rewrite the expression.

$x = \frac{π \cdot 6 - π}{3}$

Step 8.2.2.1.3

Simplify the numerator.

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Step 8.2.2.1.3.1

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{3}$

Step 8.2.2.1.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{3}$

Step 9

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

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

$\frac{2 π}{| \frac{1}{2} |}$

Step 9.3

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

$\frac{2 π}{\frac{1}{2}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 9.5

Multiply $2$ by $2$ .

$4 π$

Step 10

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

$x = \frac{π}{3} + 4 π n, \frac{5 π}{3} + 4 π n$ , for any integer $n$