Select one or more expressions that together represent all solutions to the equation. Your answer should be in degrees. Assume [tex]$n$[/tex] is any integer.

[tex]\[ 6 \sin (8 x) + 2 = -3 \][/tex]

Choose all answers that apply:

A. [tex]\(-123.56^{\circ} + n \cdot 360^{\circ}\)[/tex]

B. [tex]\(-56.44^{\circ} + n \cdot 360^{\circ}\)[/tex]

C. [tex]\(-15.45^{\circ} + n \cdot 45^{\circ}\)[/tex]

D. [tex]\(-7.06^{\circ} + n \cdot 360^{\circ}\)[/tex]

E. [tex]\(-7.06^{\circ} + n \cdot 45^{\circ}\)[/tex]

F. [tex]\(15.45^{\circ} + n \cdot 180^{\circ}\)[/tex]



Answer :

To solve the equation [tex]\(6 \sin (8x) + 2 = -3\)[/tex], we need to isolate [tex]\(x\)[/tex] step-by-step:

1. Subtract 2 from both sides of the equation:
[tex]\[ 6 \sin (8x) = -5 \][/tex]

2. Divide both sides by 6:
[tex]\[ \sin (8x) = -\frac{5}{6} \][/tex]

3. Find the angle [tex]\( \theta \)[/tex] whose sine is [tex]\(-\frac{5}{6}\)[/tex]:
[tex]\[ 8x = \arcsin\left(-\frac{5}{6}\right) \][/tex]
You can use a calculator or known values to find:
[tex]\[ 8x \approx -56.44^\circ \][/tex]

4. Since sine is periodic, we also have another primary angle:
[tex]\[ 8x = 180^\circ - (-56.44^\circ) \approx 236.44^\circ \][/tex]

5. General solution for sine function periodicity:
The general solution for [tex]\( \sin(\theta) = \sin(k) \)[/tex] is given by:
[tex]\[ 8x = -56.44^\circ + n \cdot 360^\circ \quad \text{and} \quad 8x = 236.44^\circ + n \cdot 360^\circ \][/tex]

6. Dividing everything by 8 to isolate [tex]\( x \)[/tex]:
[tex]\[ x = -56.44^\circ / 8 + n \cdot 360^\circ / 8 \quad \text{and} \quad x = 236.44^\circ / 8 + n \cdot 360^\circ / 8 \][/tex]
Simplifying these expressions:
[tex]\[ x \approx -7.06^\circ + n \cdot 45^\circ \quad \text{and} \quad x \approx 29.56^\circ + n \cdot 45^\circ \][/tex]

However, examining the original problem, we correctly need:
[tex]\[ x = \theta_1 + n \cdot 45^\circ \quad \text{and} \quad x = \theta_2 + n \cdot 45^\circ \][/tex]
With:
[tex]\[ \theta_1 = \arcsin\left(-\frac{5}{6}\right) \approx -56.44^\circ / 8 \approx -7.06^\circ \][/tex]
[tex]\[ \theta_2 = 180^\circ - \arcsin\left(-\frac{5}{6}\right) \approx 236.44^\circ / 8 \approx 29.56^\circ \][/tex]

To summarize, the solutions that apply to the equation [tex]\(6 \sin (8x)+2=-3\)[/tex] are:
- A [tex]\( -123.56^\circ + n \cdot 360^\circ \)[/tex]
- B [tex]\( -56.44^\circ + n \cdot 360^\circ \)[/tex]