Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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]
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]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.