Trigonometry Examples

$\sin (\frac{3 n}{2}) = 1$

Step 1

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

$\frac{3 n}{2} = \arcsin (1)$

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{3 n}{2} = \frac{π}{2}$

Step 3

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

$3 n = π$

Step 4

Divide each term in $3 n = π$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 n = π$ by $3$ .

$\frac{3 n}{3} = \frac{π}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 n}{3} = \frac{π}{3}$

Step 4.2.1.2

Divide $n$ by $1$ .

$n = \frac{π}{3}$

Step 5

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.

$\frac{3 n}{2} = π - \frac{π}{2}$

Step 6

Solve for $n$ .

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Step 6.1

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

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

Step 6.2

Simplify both sides of the equation.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

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

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Step 6.2.1.1.1

Cancel the common factor of $2$ .

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Step 6.2.1.1.1.1

Cancel the common factor.

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

Step 6.2.1.1.1.2

Rewrite the expression.

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

Step 6.2.1.1.2

Cancel the common factor of $3$ .

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Step 6.2.1.1.2.1

Factor $3$ out of $3 n$ .

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

Step 6.2.1.1.2.2

Cancel the common factor.

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

Step 6.2.1.1.2.3

Rewrite the expression.

$n = \frac{2}{3} (π - \frac{π}{2})$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $\frac{2}{3} (π - \frac{π}{2})$ .

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Step 6.2.2.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.2.2.1.2

Simplify terms.

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Step 6.2.2.1.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 6.2.2.1.2.2

Combine the numerators over the common denominator.

$n = \frac{2}{3} \cdot \frac{π \cdot 2 - π}{2}$

Step 6.2.2.1.2.3

Cancel the common factor of $2$ .

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Step 6.2.2.1.2.3.1

Cancel the common factor.

$n = \frac{2}{3} \cdot \frac{π \cdot 2 - π}{2}$

Step 6.2.2.1.2.3.2

Rewrite the expression.

$n = \frac{1}{3} (π \cdot 2 - π)$

Step 6.2.2.1.3

Move $2$ to the left of $π$ .

$n = \frac{1}{3} (2 π - π)$

Step 6.2.2.1.4

Simplify terms.

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Step 6.2.2.1.4.1

Subtract $π$ from $2 π$ .

$n = \frac{1}{3} π$

Step 6.2.2.1.4.2

Combine $\frac{1}{3}$ and $π$ .

$n = \frac{π}{3}$

Step 7

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

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| \frac{3}{2} |}$

Step 7.3

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

$\frac{2 π}{\frac{3}{2}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{3}$

Step 7.5

Multiply $2 π \frac{2}{3}$ .

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Step 7.5.1

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

$\frac{2 \cdot 2}{3} π$

Step 7.5.2

Multiply $2$ by $2$ .

$\frac{4}{3} π$

Step 7.5.3

Combine $\frac{4}{3}$ and $π$ .

$\frac{4 π}{3}$

Step 8

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

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