To solve the equation [tex]\( 3 \tan(5t) = 2 \)[/tex] in the interval [tex]\( 0 \leq t \leq \frac{\pi}{2} \)[/tex] radians, we can break it down into the following steps:
1. Rearrange the Equation:
First, isolate the [tex]\(\tan\)[/tex]-term:
[tex]\[
\tan(5t) = \frac{2}{3}
\][/tex]
2. Solve for the angle inside the tangent function:
We need to find [tex]\( 5t \)[/tex] such that the tangent of [tex]\(5t\)[/tex] is [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
5t = \tan^{-1}\left(\frac{2}{3}\right)
\][/tex]
Let [tex]\(\alpha = \tan^{-1}\left(\frac{2}{3}\right)\)[/tex].
3. Compute [tex]\(\alpha\)[/tex]:
We can get the approximate value of [tex]\(\alpha\)[/tex]:
[tex]\[
\alpha \approx 0.588
\][/tex]
4. Solve for [tex]\( t \)[/tex] by dividing by 5:
[tex]\[
t = \frac{\alpha}{5}
\][/tex]
So,
[tex]\[
t \approx \frac{0.588}{5} \approx 0.1176
\][/tex]
Because the tangent function is periodic, we need to consider additional solutions within the interval [tex]\(0 \leq t \leq \frac{\pi}{2}\)[/tex]. The general solution for [tex]\(5t = \alpha + n\pi\)[/tex] is given by:
[tex]\[
t = \frac{\alpha + n\pi}{5}
\][/tex]
where [tex]\( n \)[/tex] is an integer. We need to find all values of [tex]\( t \)[/tex] that satisfy this within the given interval.
5. Consider additional multiples of [tex]\(\pi\)[/tex]:
Given the range for [tex]\( t \)[/tex]:
[tex]\[
0 \leq t \leq \frac{\pi}{2}
\][/tex]
This range implies:
[tex]\[
0 \leq 5t \leq \frac{5\pi}{2}
\][/tex]
6. Find all valid [tex]\( t \)[/tex]:
We calculate the values for [tex]\( 5t \)[/tex]:
[tex]\[
5t_0 = \alpha \approx 0.588
\][/tex]
[tex]\[
5t_1 = \alpha + \pi \approx 0.588 + 3.142 \approx 3.730
\][/tex]
[tex]\[
5t_2 = \alpha + 2\pi \approx 0.588 + 6.284 \approx 6.872
\][/tex]
Thus, the equivalent [tex]\( t \)[/tex] values are:
[tex]\[
t_0 = \frac{0.588}{5} \approx 0.1176
\][/tex]
[tex]\[
t_1 = \frac{3.730}{5} \approx 0.746
\][/tex]
[tex]\[
t_2 = \frac{6.872}{5} \approx 1.374
\][/tex]
Therefore, the solutions in the interval [tex]\(0 \leq t \leq \frac{\pi}{2}\)[/tex] are [tex]\( t \approx 0.1176 \)[/tex], [tex]\(t \approx 0.7459\)[/tex], and [tex]\( t \approx 1.3742\)[/tex].
The correct answer is:
c) [tex]\(0.1176, 0.7459, 1.3742\)[/tex]
