Trigonometry Examples

$\sin (\frac{x}{6}) = 0$

Step 1

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

$\frac{x}{6} = \arcsin (0)$

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{x}{6} = 0$

Step 3

Set the numerator equal to zero.

$x = 0$

Step 4

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.

$\frac{x}{6} = π - 0$

Step 5

Solve for $x$ .

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Step 5.1

Multiply both sides of the equation by $6$ .

$6 \frac{x}{6} = 6 (π - 0)$

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Cancel the common factor of $6$ .

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Step 5.2.1.1.1

Cancel the common factor.

$6 \frac{x}{6} = 6 (π - 0)$

Step 5.2.1.1.2

Rewrite the expression.

$x = 6 (π - 0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Subtract $0$ from $π$ .

$x = 6 π$

Step 6

Find the period of $\sin (\frac{x}{6})$ .

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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 $\frac{1}{6}$ in the formula for period.

$\frac{2 π}{| \frac{1}{6} |}$

Step 6.3

$\frac{1}{6}$ is approximately $0.1\overline{6}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{6}}$

Step 6.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 6$

Step 6.5

Multiply $6$ by $2$ .

$12 π$

Step 7

The period of the $\sin (\frac{x}{6})$ function is $12 π$ so values will repeat every $12 π$ radians in both directions.

$x = 12 π n, 6 π + 12 π n$ , for any integer $n$

Step 8

Consolidate the answers.

$x = 6 π n$ , for any integer $n$