Trigonometry Examples

$2 \sin (x + \frac{π}{6}) + 1 = 0$

Step 1

Subtract $1$ from both sides of the equation.

$2 \sin (x + \frac{π}{6}) = - 1$

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

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

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

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Simplify the right side.

Tap for more steps...

Step 4.1

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

$x + \frac{π}{6} = - \frac{π}{6}$

Step 5

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

Tap for more steps...

Step 5.1

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

$x = - \frac{π}{6} - \frac{π}{6}$

Step 5.2

Combine the numerators over the common denominator.

$x = \frac{- π - π}{6}$

Step 5.3

Subtract $π$ from $- π$ .

$x = \frac{- 2 π}{6}$

Step 5.4

Cancel the common factor of $- 2$ and $6$ .

Tap for more steps...

Step 5.4.1

Factor $2$ out of $- 2 π$ .

$x = \frac{2 (- π)}{6}$

Step 5.4.2

Cancel the common factors.

Tap for more steps...

Step 5.4.2.1

Factor $2$ out of $6$ .

$x = \frac{2 (- π)}{2 (3)}$

Step 5.4.2.2

Cancel the common factor.

$x = \frac{2 (- π)}{2 \cdot 3}$

Step 5.4.2.3

Rewrite the expression.

$x = \frac{- π}{3}$

Step 5.5

Move the negative in front of the fraction.

$x = - \frac{π}{3}$

Step 6

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x + \frac{π}{6} = 2 π + \frac{π}{6} + π$

Step 7

Simplify the expression to find the second solution.

Tap for more steps...

Step 7.1

Subtract $2 π$ from $2 π + \frac{π}{6} + π$ .

$x + \frac{π}{6} = 2 π + \frac{π}{6} + π - 2 π$

Step 7.2

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{6} + π$ .

$x + \frac{π}{6} = \frac{7 π}{6}$

Step 7.3

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

Tap for more steps...

Step 7.3.1

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

$x = \frac{7 π}{6} - \frac{π}{6}$

Step 7.3.2

Combine the numerators over the common denominator.

$x = \frac{7 π - π}{6}$

Step 7.3.3

Subtract $π$ from $7 π$ .

$x = \frac{6 π}{6}$

Step 7.3.4

Cancel the common factor of $6$ .

Tap for more steps...

Step 7.3.4.1

Cancel the common factor.

$x = \frac{6 π}{6}$

Step 7.3.4.2

Divide $π$ by $1$ .

$x = π$

Step 8

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

Tap for more steps...

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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 8.4

Divide $2 π$ by $1$ .

$2 π$

Step 9

Add $2 π$ to every negative angle to get positive angles.

Tap for more steps...

Step 9.1

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

$- \frac{π}{3} + 2 π$

Step 9.2

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

$2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 9.3

Combine fractions.

Tap for more steps...

Step 9.3.1

Combine $2 π$ and $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 9.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - π}{3}$

Step 9.4

Simplify the numerator.

Tap for more steps...

Step 9.4.1

Multiply $3$ by $2$ .

$\frac{6 π - π}{3}$

Step 9.4.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{3}$

Step 9.5

List the new angles.

$x = \frac{5 π}{3}$

Step 10

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

$x = π + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$