Trigonometry Examples

$0 = 1 - 2 \sin (- 3 x + \frac{π}{6})$

Step 1

Rewrite the equation as $1 - 2 \sin (- 3 x + \frac{π}{6}) = 0$ .

$1 - 2 \sin (- 3 x + \frac{π}{6}) = 0$

Step 2

Subtract $1$ from both sides of the equation.

$- 2 \sin (- 3 x + \frac{π}{6}) = - 1$

Step 3

Divide each term in $- 2 \sin (- 3 x + \frac{π}{6}) = - 1$ by $- 2$ and simplify.

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Step 3.1

Divide each term in $- 2 \sin (- 3 x + \frac{π}{6}) = - 1$ by $- 2$ .

$\frac{- 2 \sin (- 3 x + \frac{π}{6})}{- 2} = \frac{- 1}{- 2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 2 \sin (- 3 x + \frac{π}{6})}{- 2} = \frac{- 1}{- 2}$

Step 3.2.1.2

Divide $\sin (- 3 x + \frac{π}{6})$ by $1$ .

$\sin (- 3 x + \frac{π}{6}) = \frac{- 1}{- 2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Dividing two negative values results in a positive value.

$\sin (- 3 x + \frac{π}{6}) = \frac{1}{2}$

Step 4

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

$- 3 x + \frac{π}{6} = \arcsin (\frac{1}{2})$

Step 5

Simplify the right side.

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Step 5.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$- 3 x + \frac{π}{6} = \frac{π}{6}$

Step 6

Move all terms not containing $x$ to the right side of the equation.

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Step 6.1

Subtract $\frac{π}{6}$ from both sides of the equation.

$- 3 x = \frac{π}{6} - \frac{π}{6}$

Step 6.2

Combine the numerators over the common denominator.

$- 3 x = \frac{π - π}{6}$

Step 6.3

Subtract $π$ from $π$ .

$- 3 x = \frac{0}{6}$

Step 6.4

Divide $0$ by $6$ .

$- 3 x = 0$

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 3$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $x$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $0$ by $- 3$ .

$x = 0$

Step 8

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.

$- 3 x + \frac{π}{6} = π - \frac{π}{6}$

Step 9

Solve for $x$ .

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Step 9.1

Simplify $π - \frac{π}{6}$ .

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Step 9.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- 3 x + \frac{π}{6} = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 9.1.2

Combine fractions.

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Step 9.1.2.1

Combine $π$ and $\frac{6}{6}$ .

$- 3 x + \frac{π}{6} = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 9.1.2.2

Combine the numerators over the common denominator.

$- 3 x + \frac{π}{6} = \frac{π \cdot 6 - π}{6}$

Step 9.1.3

Simplify the numerator.

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Step 9.1.3.1

Move $6$ to the left of $π$ .

$- 3 x + \frac{π}{6} = \frac{6 \cdot π - π}{6}$

Step 9.1.3.2

Subtract $π$ from $6 π$ .

$- 3 x + \frac{π}{6} = \frac{5 π}{6}$

Step 9.2

Move all terms not containing $x$ to the right side of the equation.

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Step 9.2.1

Subtract $\frac{π}{6}$ from both sides of the equation.

$- 3 x = \frac{5 π}{6} - \frac{π}{6}$

Step 9.2.2

Combine the numerators over the common denominator.

$- 3 x = \frac{5 π - π}{6}$

Step 9.2.3

Subtract $π$ from $5 π$ .

$- 3 x = \frac{4 π}{6}$

Step 9.2.4

Cancel the common factor of $4$ and $6$ .

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Step 9.2.4.1

Factor $2$ out of $4 π$ .

$- 3 x = \frac{2 (2 π)}{6}$

Step 9.2.4.2

Cancel the common factors.

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Step 9.2.4.2.1

Factor $2$ out of $6$ .

$- 3 x = \frac{2 (2 π)}{2 (3)}$

Step 9.2.4.2.2

Cancel the common factor.

$- 3 x = \frac{2 (2 π)}{2 \cdot 3}$

Step 9.2.4.2.3

Rewrite the expression.

$- 3 x = \frac{2 π}{3}$

Step 9.3

Divide each term in $- 3 x = \frac{2 π}{3}$ by $- 3$ and simplify.

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Step 9.3.1

Divide each term in $- 3 x = \frac{2 π}{3}$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{\frac{2 π}{3}}{- 3}$

Step 9.3.2

Simplify the left side.

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Step 9.3.2.1

Cancel the common factor of $- 3$ .

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Step 9.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{\frac{2 π}{3}}{- 3}$

Step 9.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{2 π}{3}}{- 3}$

Step 9.3.3

Simplify the right side.

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Step 9.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{2 π}{3} \cdot \frac{1}{- 3}$

Step 9.3.3.2

Move the negative in front of the fraction.

$x = \frac{2 π}{3} (- \frac{1}{3})$

Step 9.3.3.3

Multiply $\frac{2 π}{3} (- \frac{1}{3})$ .

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Step 9.3.3.3.1

Multiply $\frac{2 π}{3}$ by $\frac{1}{3}$ .

$x = - \frac{2 π}{3 \cdot 3}$

Step 9.3.3.3.2

Multiply $3$ by $3$ .

$x = - \frac{2 π}{9}$

Step 10

Find the period of $\sin (- 3 x + \frac{π}{6})$ .

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Step 10.1

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

$\frac{2 π}{| b |}$

Step 10.2

Replace $b$ with $- 3$ in the formula for period.

$\frac{2 π}{| - 3 |}$

Step 10.3

The absolute value is the distance between a number and zero. The distance between $- 3$ and $0$ is $3$ .

$\frac{2 π}{3}$

Step 11

Add $\frac{2 π}{3}$ to every negative angle to get positive angles.

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Step 11.1

Add $\frac{2 π}{3}$ to $- \frac{2 π}{9}$ to find the positive angle.

$- \frac{2 π}{9} + \frac{2 π}{3}$

Step 11.2

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 π}{3} \cdot \frac{3}{3} - \frac{2 π}{9}$

Step 11.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.3.1

Multiply $\frac{2 π}{3}$ by $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{3 \cdot 3} - \frac{2 π}{9}$

Step 11.3.2

Multiply $3$ by $3$ .

$\frac{2 π \cdot 3}{9} - \frac{2 π}{9}$

Step 11.4

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - 2 π}{9}$

Step 11.5

Simplify the numerator.

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Step 11.5.1

Multiply $3$ by $2$ .

$\frac{6 π - 2 π}{9}$

Step 11.5.2

Subtract $2 π$ from $6 π$ .

$\frac{4 π}{9}$

Step 11.6

List the new angles.

$x = \frac{4 π}{9}$

Step 12

The period of the $\sin (- 3 x + \frac{π}{6})$ function is $\frac{2 π}{3}$ so values will repeat every $\frac{2 π}{3}$ radians in both directions.

$x = \frac{2 π n}{3}, \frac{4 π}{9} + \frac{2 π n}{3}$ , for any integer $n$