Trigonometry Examples

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

Step 1

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

$\frac{3}{2} x = \arcsin (0)$

Step 2

Simplify the left side.

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Step 2.1

Combine $\frac{3}{2}$ and $x$ .

$\frac{3 x}{2} = \arcsin (0)$

Step 3

Simplify the right side.

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Step 3.1

The exact value of $\arcsin (0)$ is $0$ .

$\frac{3 x}{2} = 0$

Step 4

Set the numerator equal to zero.

$3 x = 0$

Step 5

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $3$ .

$x = 0$

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.

$\frac{3 x}{2} = π - 0$

Step 7

Solve for $x$ .

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Step 7.1

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} (π - 0)$

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

Simplify $\frac{2}{3} \cdot \frac{3 x}{2}$ .

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Step 7.2.1.1.1

Cancel the common factor of $2$ .

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Step 7.2.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} (π - 0)$

Step 7.2.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{2}{3} (π - 0)$

Step 7.2.1.1.2

Cancel the common factor of $3$ .

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Step 7.2.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{2}{3} (π - 0)$

Step 7.2.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{2}{3} (π - 0)$

Step 7.2.1.1.2.3

Rewrite the expression.

$x = \frac{2}{3} (π - 0)$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $\frac{2}{3} (π - 0)$ .

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Step 7.2.2.1.1

Subtract $0$ from $π$ .

$x = \frac{2}{3} π$

Step 7.2.2.1.2

Combine $\frac{2}{3}$ and $π$ .

$x = \frac{2 π}{3}$

Step 8

Find the period of $\sin (\frac{3}{2} x)$ .

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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 $\frac{3}{2}$ in the formula for period.

$\frac{2 π}{| \frac{3}{2} |}$

Step 8.3

$\frac{3}{2}$ is approximately $1.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3}{2}}$

Step 8.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{3}$

Step 8.5

Multiply $2 π \frac{2}{3}$ .

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Step 8.5.1

Combine $\frac{2}{3}$ and $2$ .

$\frac{2 \cdot 2}{3} π$

Step 8.5.2

Multiply $2$ by $2$ .

$\frac{4}{3} π$

Step 8.5.3

Combine $\frac{4}{3}$ and $π$ .

$\frac{4 π}{3}$

Step 9

The period of the $\sin (\frac{3}{2} x)$ function is $\frac{4 π}{3}$ so values will repeat every $\frac{4 π}{3}$ radians in both directions.

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

Step 10

Consolidate the answers.

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