To solve the equation [tex]\( -\frac{3}{2} + \sin t = -2 \)[/tex] within the interval [tex]\( 0 \leq t < 4\pi \)[/tex], follow these steps:
1. Isolate [tex]\(\sin t\)[/tex]: Start by isolating the sine function
[tex]\[
-\frac{3}{2} + \sin t = -2.
\][/tex]
Add [tex]\(\frac{3}{2}\)[/tex] to both sides to isolate [tex]\(\sin t\)[/tex]:
[tex]\[
\sin t = -2 + \frac{3}{2}.
\][/tex]
2. Simplify the equation: Simplify the right-hand side:
[tex]\[
\sin t = -0.5.
\][/tex]
3. Find the values of [tex]\(t\)[/tex]: We need to find [tex]\(t\)[/tex] such that [tex]\(\sin t = -0.5\)[/tex] within the interval [tex]\( 0 \leq t < 4\pi \)[/tex].
- In the interval [tex]\( 0 \leq t < 2\pi \)[/tex], [tex]\(\sin t = -0.5\)[/tex] at two points:
[tex]\[
t_1 = \frac{7\pi}{6} \quad \text{and} \quad t_2 = \frac{11\pi}{6}.
\][/tex]
- Since the sine function is periodic with a period of [tex]\(2\pi\)[/tex], we can find additional solutions by adding multiples of [tex]\(2\pi\)[/tex]. We are interested in finding [tex]\(t\)[/tex] within the interval [tex]\(0 \leq t < 4\pi\)[/tex].
Add [tex]\(2\pi\)[/tex] to each of these solutions:
[tex]\[
t_3 = \frac{7\pi}{6} + 2\pi = \frac{7\pi}{6} + \frac{12\pi}{6} = \frac{19\pi}{6},
\][/tex]
[tex]\[
t_4 = \frac{11\pi}{6} + 2\pi = \frac{11\pi}{6} + \frac{12\pi}{6} = \frac{23\pi}{6}.
\][/tex]
4. List the solutions: Collect all the solutions that fall within the interval [tex]\( 0 \leq t < 4\pi \)[/tex]:
[tex]\[
t = \left\{ \frac{7\pi}{6}, \frac{11\pi}{6}, \frac{19\pi}{6}, \frac{23\pi}{6} \right\}.
\][/tex]
In decimal form, these values are approximately:
[tex]\[
t = \left\{ 3.67, 5.76, 9.95, 12.04 \right\}.
\][/tex]
Therefore, the solutions to the equation [tex]\( -\frac{3}{2} + \sin t = -2 \)[/tex] within the interval [tex]\( 0 \leq t < 4\pi \)[/tex] are:
[tex]\[
t = \boxed{ \left\{ 3.67, 5.76, 9.95, 12.04 \right\} }
\][/tex]
