Trigonometry Examples

$6 \sin (8 x) + 2 = - 3$

Step 1

Move all terms not containing $\sin (8 x)$ to the right side of the equation.

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Step 1.1

Subtract $2$ from both sides of the equation.

$6 \sin (8 x) = - 3 - 2$

Step 1.2

Subtract $2$ from $- 3$ .

$6 \sin (8 x) = - 5$

Step 2

Divide each term in $6 \sin (8 x) = - 5$ by $6$ and simplify.

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Step 2.1

Divide each term in $6 \sin (8 x) = - 5$ by $6$ .

$\frac{6 \sin (8 x)}{6} = \frac{- 5}{6}$

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.

$\frac{6 \sin (8 x)}{6} = \frac{- 5}{6}$

Step 2.2.1.2

Divide $\sin (8 x)$ by $1$ .

$\sin (8 x) = \frac{- 5}{6}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$\sin (8 x) = - \frac{5}{6}$

Step 3

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

$8 x = \arcsin (- \frac{5}{6})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arcsin (- \frac{5}{6})$ .

$8 x = - 56.44269023$

Step 5

Divide each term in $8 x = - 56.44269023$ by $8$ and simplify.

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Step 5.1

Divide each term in $8 x = - 56.44269023$ by $8$ .

$\frac{8 x}{8} = \frac{- 56.44269023}{8}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $8$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 56.44269023}{8}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 56.44269023}{8}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $- 56.44269023$ by $8$ .

$x = - 7.05533627$

Step 6

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $360$ , to find a reference angle. Next, add this reference angle to $180$ to find the solution in the third quadrant.

$8 x = 360 + 56.44269023 + 180$

Step 7

Simplify the expression to find the second solution.

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Step 7.1

Subtract $360 °$ from $360 + 56.44269023 + 180 °$ .

$8 x = 360 + 56.44269023 + 180 ° - 360 °$

Step 7.2

The resulting angle of $236.44269023 °$ is positive, less than $360 °$ , and coterminal with $360 + 56.44269023 + 180$ .

$8 x = 236.44269023 °$

Step 7.3

Divide each term in $8 x = 236.44269023 °$ by $8$ and simplify.

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Step 7.3.1

Divide each term in $8 x = 236.44269023 °$ by $8$ .

$\frac{8 x}{8} = \frac{236.44269023 °}{8}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $8$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{236.44269023 °}{8}$

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{236.44269023 °}{8}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Divide $236.44269023 °$ by $8$ .

$x = 29.55533627$

Step 8

Find the period of $\sin (8 x)$ .

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Step 8.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 8.2

Replace $b$ with $8$ in the formula for period.

$\frac{360}{| 8 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$\frac{360}{8}$

Step 8.4

Divide $360$ by $8$ .

$45$

Step 9

Add $45$ to every negative angle to get positive angles.

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Step 9.1

Add $45$ to $- 7.05533627$ to find the positive angle.

$- 7.05533627 + 45$

Step 9.2

Subtract $7.05533627$ from $45$ .

$37.94466372$

Step 9.3

List the new angles.

$x = 37.94466372$

Step 10

The period of the $\sin (8 x)$ function is $45$ so values will repeat every $45$ degrees in both directions.

$x = 29.55533627 + 45 n, 37.94466372 + 45 n$ , for any integer $n$