Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{x}{2} = - \frac{π}{2}$

Step 3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = - π$

Step 4

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.

$\frac{x}{2} = 2 π + \frac{π}{2} + π$

Step 5

Simplify the expression to find the second solution.

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Step 5.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

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

Step 5.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$\frac{x}{2} = \frac{3 π}{2}$

Step 5.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = 3 π$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| \frac{1}{2} |}$

Step 6.3

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

$\frac{2 π}{\frac{1}{2}}$

Step 6.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 6.5

Multiply $2$ by $2$ .

$4 π$

Step 7

Add $4 π$ to every negative angle to get positive angles.

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Step 7.1

Add $4 π$ to $- π$ to find the positive angle.

$- π + 4 π$

Step 7.2

Subtract $π$ from $4 π$ .

$3 π$

Step 7.3

List the new angles.

$x = 3 π$

Step 8

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

$x = 3 π + 4 π n, 3 π + 4 π n$ , for any integer $n$

Step 9

Consolidate the answers.

$x = 3 π + 4 π n$ , for any integer $n$