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Sagot :
Answer:
Step-by-step explanation:
Hey there! To solve the equation \( -6 \sin(4\theta) = 3 \sqrt{34} \) for \( 0 \leq \theta < 2\pi \), we need to isolate \( \sin(4\theta) \) first.
Divide both sides by -6:
\( \sin(4\theta) = -\frac{3 \sqrt{34}}{6} = -\frac{\sqrt{34}}{2} \)
Now, to find the values of \( \theta \) that satisfy this equation, we need to look at the unit circle and find where the sine function equals \( -\frac{\sqrt{34}}{2} \).
The reference angle for \( \sin^{-1}(\frac{\sqrt{34}}{2}) \) is \( \frac{\pi}{6} \), which corresponds to \( \frac{\pi}{6} \) and \( \frac{5\pi}{6} \) in the second and third quadrants, respectively.
Since we are looking for \( 0 \leq \theta < 2\pi \), the solutions are \( \frac{7\pi}{6} \) and \( \frac{11\pi}{6} \).
So, the answers are \( \theta = \frac{7\pi}{6} \) and \( \theta = \frac{11\pi}{6} \).
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