Solve for x 6sin(8x)+2=-3

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

Step 1

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

Subtract $2$ from both sides of the equation.

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

Subtract $2$ from $- 3$ .

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

Step 2

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

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

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

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

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

Simplify the right side.

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.

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

$8 x = - 0.98511078$

Step 5

Divide each term in $8 x = - 0.98511078$ by $8$ and simplify.

Divide each term in $8 x = - 0.98511078$ by $8$ .

$\frac{8 x}{8} = \frac{- 0.98511078}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 0.98511078}{8}$

Divide $x$ by $1$ .

$x = \frac{- 0.98511078}{8}$

Simplify the right side.

Divide $- 0.98511078$ by $8$ .

$x = - 0.12313884$

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.

$8 x = 2 (3.14159265) + 0.98511078 + 3.14159265$

Step 7

Simplify the expression to find the second solution.

Subtract $2 π$ from $2 (3.14159265) + 0.98511078 + 3.14159265$ .

$8 x = 2 (3.14159265) + 0.98511078 + 3.14159265 - 2 π$

The resulting angle of $4.12670343$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.98511078 + 3.14159265$ .

$8 x = 4.12670343$

Divide each term in $8 x = 4.12670343$ by $8$ and simplify.

Divide each term in $8 x = 4.12670343$ by $8$ .

$\frac{8 x}{8} = \frac{4.12670343}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{4.12670343}{8}$

Divide $x$ by $1$ .

$x = \frac{4.12670343}{8}$

Simplify the right side.

Divide $4.12670343$ by $8$ .

$x = 0.51583792$

Step 8

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 8 |}$

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

$\frac{2 π}{8}$

Cancel the common factor of $2$ and $8$ .

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$\frac{2 π}{2 \cdot 4}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 4}$

Rewrite the expression.

$\frac{π}{4}$

Step 9

Add $\frac{π}{4}$ to every negative angle to get positive angles.

Add $\frac{π}{4}$ to $- 0.12313884$ to find the positive angle.

$- 0.12313884 + \frac{π}{4}$

Subtract $0.12313884$ from $\frac{π}{4}$ .

$0.66225931$

List the new angles.

$x = 0.66225931$

Step 10

The period of the $\sin (8 x)$ function is $\frac{π}{4}$ so values will repeat every $\frac{π}{4}$ radians in both directions.

$x = 0.51583792 + \frac{π n}{4}, 0.66225931 + \frac{π n}{4}$ , for any integer $n$

Solve for x 6sin(8x)+2=-3

Related Questions

Support LCMGCF.com by disabling your adblocker.