Trigonometry Examples

$6 \sin (x) + 2 = 0$

Step 1

Subtract $2$ from both sides of the equation.

$6 \sin (x) = - 2$

Step 2

Divide each term in $6 \sin (x) = - 2$ by $6$ and simplify.

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Step 2.1

Divide each term in $6 \sin (x) = - 2$ by $6$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $6$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $\sin (x)$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Factor $2$ out of $- 2$ .

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

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $6$ .

$\sin (x) = \frac{2 \cdot - 1}{2 \cdot 3}$

Step 2.3.1.2.2

Cancel the common factor.

$\sin (x) = \frac{2 \cdot - 1}{2 \cdot 3}$

Step 2.3.1.2.3

Rewrite the expression.

$\sin (x) = \frac{- 1}{3}$

Step 2.3.2

Move the negative in front of the fraction.

$\sin (x) = - \frac{1}{3}$

Step 3

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

$x = \arcsin (- \frac{1}{3})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arcsin (- \frac{1}{3})$ .

$x = - 0.3398369$

Step 5

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.

$x = 2 (3.14159265) + 0.3398369 + 3.14159265$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Subtract $2 π$ from $2 (3.14159265) + 0.3398369 + 3.14159265$ .

$x = 2 (3.14159265) + 0.3398369 + 3.14159265 - 2 π$

Step 6.2

The resulting angle of $3.48142956$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.3398369 + 3.14159265$ .

$x = 3.48142956$

Step 7

Find the period of $\sin (x)$ .

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 7.3

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

$\frac{2 π}{1}$

Step 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

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

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Step 8.1

Add $2 π$ to $- 0.3398369$ to find the positive angle.

$- 0.3398369 + 2 π$

Step 8.2

Subtract $0.3398369$ from $2 π$ .

$5.94334839$

Step 8.3

List the new angles.

$x = 5.94334839$

Step 9

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

$x = 3.48142956 + 2 π n, 5.94334839 + 2 π n$ , for any integer $n$