To find \(\sin(2t)\) given \(\pi < t < \frac{3\pi}{2}\) and \(\sin(t) = -\frac{3}{4}\), we can use the double-angle formula for sine. Let's go through the steps one-by-one:
1. Identify Given Information:
- \(\sin(t) = -\frac{3}{4}\)
- The angle \(t\) is such that \(\pi < t < \frac{3\pi}{2}\) which places \(t\) in the third quadrant.
2. Calculate \(\cos(t)\):
- Use the Pythagorean identity: \(\sin^2(t) + \cos^2(t) = 1\)
- Substituting \(\sin(t)\):
[tex]\[
\left(-\frac{3}{4}\right)^2 + \cos^2(t) = 1
\][/tex]
- Simplify:
[tex]\[
\frac{9}{16} + \cos^2(t) = 1
\][/tex]
- Solve for \(\cos^2(t)\):
[tex]\[
\cos^2(t) = 1 - \frac{9}{16}
\][/tex]
[tex]\[
\cos^2(t) = \frac{16}{16} - \frac{9}{16}
\][/tex]
[tex]\[
\cos^2(t) = \frac{7}{16}
\][/tex]
- Since \(t\) is in the third quadrant, \(\cos(t)\) is negative:
[tex]\[
\cos(t) = -\sqrt{\frac{7}{16}} = -\frac{\sqrt{7}}{4}
\][/tex]
3. Use the Double-Angle Formula:
- The double-angle formula for sine is:
[tex]\[
\sin(2t) = 2 \sin(t) \cos(t)
\][/tex]
- Substitute the values:
[tex]\[
\sin(2t) = 2 \left(-\frac{3}{4}\right) \left(-\frac{\sqrt{7}}{4}\right)
\][/tex]
- Simplify:
[tex]\[
\sin(2t) = 2 \cdot \frac{3\sqrt{7}}{16}
\][/tex]
[tex]\[
\sin(2t) = \frac{6\sqrt{7}}{16}
\][/tex]
[tex]\[
\sin(2t) = \frac{3\sqrt{7}}{8}
\][/tex]
Therefore, the exact value of \(\sin(2t)\) is:
[tex]\[
\sin(2t) = \frac{3\sqrt{7}}{8}
\][/tex]
