Trigonometry Examples

$4 \sin (\frac{π}{6} x) = 3$

Step 1

Divide each term in $4 \sin (\frac{π}{6} x) = 3$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 \sin (\frac{π}{6} x) = 3$ by $4$ .

$\frac{4 \sin (\frac{π}{6} x)}{4} = \frac{3}{4}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $4$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{4 \sin (\frac{π}{6} x)}{4} = \frac{3}{4}$

Step 1.2.1.2

Divide $\sin (\frac{π}{6} x)$ by $1$ .

$\sin (\frac{π}{6} x) = \frac{3}{4}$

Step 2

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

$\frac{π}{6} x = \arcsin (\frac{3}{4})$

Step 3

Simplify the left side.

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Step 3.1

Combine $\frac{π}{6}$ and $x$ .

$\frac{π x}{6} = \arcsin (\frac{3}{4})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arcsin (\frac{3}{4})$ .

$\frac{π x}{6} = 0.84806207$

Step 5

Multiply both sides of the equation by $\frac{6}{π}$ .

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} \cdot 0.84806207$

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

Simplify $\frac{6}{π} \cdot \frac{π x}{6}$ .

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Step 6.1.1.1

Cancel the common factor of $6$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} \cdot 0.84806207$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{6}{π} \cdot 0.84806207$

Step 6.1.1.2

Cancel the common factor of $π$ .

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Step 6.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{6}{π} \cdot 0.84806207$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{6}{π} \cdot 0.84806207$

Step 6.1.1.2.3

Rewrite the expression.

$x = \frac{6}{π} \cdot 0.84806207$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{6}{π} \cdot 0.84806207$ .

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Step 6.2.1.1

Multiply $\frac{6}{π} \cdot 0.84806207$ .

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Step 6.2.1.1.1

Combine $\frac{6}{π}$ and $0.84806207$ .

$x = \frac{6 \cdot 0.84806207}{π}$

Step 6.2.1.1.2

Multiply $6$ by $0.84806207$ .

$x = \frac{5.08837247}{π}$

Step 6.2.1.2

Replace $π$ with an approximation.

$x = \frac{5.08837247}{3.14159265}$

Step 6.2.1.3

Divide $5.08837247$ by $3.14159265$ .

$x = 1.61967926$

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.

$\frac{π x}{6} = (3.14159265) - 0.84806207$

Step 8

Solve for $x$ .

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Step 8.1

Multiply both sides of the equation by $\frac{6}{π}$ .

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} ((3.14159265) - 0.84806207)$

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

Simplify $\frac{6}{π} \cdot \frac{π x}{6}$ .

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Step 8.2.1.1.1

Cancel the common factor of $6$ .

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Step 8.2.1.1.1.1

Cancel the common factor.

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} ((3.14159265) - 0.84806207)$

Step 8.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{6}{π} ((3.14159265) - 0.84806207)$

Step 8.2.1.1.2

Cancel the common factor of $π$ .

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Step 8.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{6}{π} ((3.14159265) - 0.84806207)$

Step 8.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{6}{π} ((3.14159265) - 0.84806207)$

Step 8.2.1.1.2.3

Rewrite the expression.

$x = \frac{6}{π} ((3.14159265) - 0.84806207)$

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

Simplify $\frac{6}{π} ((3.14159265) - 0.84806207)$ .

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Step 8.2.2.1.1

Subtract $0.84806207$ from $3.14159265$ .

$x = \frac{6}{π} \cdot 2.29353057$

Step 8.2.2.1.2

Multiply $\frac{6}{π} \cdot 2.29353057$ .

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Step 8.2.2.1.2.1

Combine $\frac{6}{π}$ and $2.29353057$ .

$x = \frac{6 \cdot 2.29353057}{π}$

Step 8.2.2.1.2.2

Multiply $6$ by $2.29353057$ .

$x = \frac{13.76118344}{π}$

Step 8.2.2.1.3

Replace $π$ with an approximation.

$x = \frac{13.76118344}{3.14159265}$

Step 8.2.2.1.4

Divide $13.76118344$ by $3.14159265$ .

$x = 4.38032073$

Step 9

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

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

$\frac{2 π}{| \frac{π}{6} |}$

Step 9.3

$\frac{π}{6}$ is approximately $0.52359877$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{6}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{6}{π}$

Step 9.5

Cancel the common factor of $π$ .

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Step 9.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{6}{π}$

Step 9.5.2

Cancel the common factor.

$π \cdot 2 \frac{6}{π}$

Step 9.5.3

Rewrite the expression.

$2 \cdot 6$

Step 9.6

Multiply $2$ by $6$ .

$12$

Step 10

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

$x = 1.61967926 + 12 n, 4.38032073 + 12 n$ , for any integer $n$