To find all solutions to the equation [tex]\( 3 \sin(5t) = 2 \)[/tex] within the interval [tex]\( 0 \leq t \leq \frac{\pi}{2} \)[/tex], we will follow these steps:
1. Isolate the sine function:
[tex]\[
\sin(5t) = \frac{2}{3}
\][/tex]
2. Find the general solution for [tex]\( \sin(5t) \)[/tex] when [tex]\( \sin(5t) = \frac{2}{3} \)[/tex]:
- To find [tex]\(5t\)[/tex], we need to use the inverse sine function (arcsine).
- Let [tex]\( x = 5t \)[/tex]. Then, [tex]\( \sin(x) = \frac{2}{3} \)[/tex].
- The primary solutions for [tex]\( x \)[/tex] in the interval [tex]\([0, \pi]\)[/tex] are: [tex]\( x = \arcsin\left(\frac{2}{3}\right) \)[/tex] and [tex]\( x = \pi - \arcsin\left(\frac{2}{3}\right) \)[/tex].
3. Calculate numeric values:
- Note that [tex]\(\arcsin\left(\frac{2}{3}\right)\)[/tex] gives us an angle in radians whose sine is [tex]\( \frac{2}{3} \)[/tex].
- Let's denote [tex]\( \arcsin\left(\frac{2}{3}\right) = \theta \)[/tex].
4. Find [tex]\( t_1 \)[/tex] and [tex]\( t_2 \)[/tex]:
Solving for [tex]\( x = 5t \)[/tex]:
- [tex]\( 5t_1 = \theta \)[/tex] [tex]\(\Rightarrow t_1 = \frac{\theta}{5} \)[/tex]
- [tex]\( 5t_2 = \pi - \theta \)[/tex] [tex]\(\Rightarrow t_2 = \frac{\pi - \theta}{5} \)[/tex]
By knowing that:
[tex]\[
\theta \approx 0.7297 \, \text{radians}
\][/tex]
- [tex]\( t_1 = \frac{0.7297}{5} \approx 0.1459 \)[/tex]
- [tex]\( t_2 = \frac{\pi - 0.7297}{5} \approx 0.4824 \)[/tex]
These give us the two solutions within the interval [tex]\( 0 \leq t \leq \frac{\pi}{2} \)[/tex].
5. Verify if there are any other solutions within the interval:
Since [tex]\( 5t \)[/tex] must be in [tex]\([0, \pi]\)[/tex]:
[tex]\[
0 \leq 5t \leq \pi
\Rightarrow 0 \leq t \leq \frac{\pi}{5}
\approx 0 \leq t \leq 0.6283
\][/tex]
Hence, in the interval [tex]\( 0 \leq t \leq \frac{\pi}{2} \)[/tex], the previously calculated [tex]\( t_1 \approx 0.1459 \)[/tex] and [tex]\( t_2 \approx 0.4824 \)[/tex] are the only solutions.
Thus, the correct answer is:
b) [tex]\( 0.1459, 0.4824 \)[/tex]
