To find the exact value of the expression [tex]\(\sin \left(-\frac{3 \pi}{8}\right)\)[/tex] using the half-angle formulas, follow these steps:
1. Identify the relevant angle for the half-angle formula:
The half-angle formula for sine is given by:
[tex]\[
\sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos(x)}{2}}
\][/tex]
Here, we need to express [tex]\(-\frac{3\pi}{8}\)[/tex] in terms of the angle [tex]\(x\)[/tex]. Note that [tex]\(\sin\left(-\theta\right) = -\sin\left(\theta\right)\)[/tex]. We can use the angle [tex]\(\frac{3\pi}{4}\)[/tex] because:
[tex]\[
-\frac{3\pi}{8} = -\frac{1}{2} \times \frac{3\pi}{4}
\][/tex]
2. Calculate [tex]\(\cos\left(\frac{3\pi}{4}\right)\)[/tex]:
The cosine of [tex]\(\frac{3\pi}{4}\)[/tex] is:
[tex]\[
\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\][/tex]
3. Apply the half-angle formula:
[tex]\[
\sin\left(-\frac{3\pi}{8}\right) = -\sin\left(\frac{3\pi}{8}\right)
\][/tex]
Since:
[tex]\[
\sin\left(\frac{3\pi}{8}\right) = \sqrt{\frac{1 - \cos\left(\frac{3\pi}{4}\right)}{2}}
\][/tex]
4. Substitute [tex]\(\cos\left(\frac{3\pi}{4}\right)\)[/tex] into the formula:
[tex]\[
\sin\left(\frac{3\pi}{8}\right) = \sqrt{\frac{1 - \left(-\frac{\sqrt{2}}{2}\right)}{2}} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}
\][/tex]
5. Simplify the expression under the square root:
[tex]\[
\sin\left(\frac{3\pi}{8}\right) = \sqrt{\frac{2 + \sqrt{2}}{4}} = \sqrt{\frac{2 + \sqrt{2}}{4}} = \frac{\sqrt{2 + \sqrt{2}}}{2}
\][/tex]
6. Account for the sign:
Since we are dealing with [tex]\(\sin\left(-\frac{3\pi}{8}\right)\)[/tex], and since [tex]\(\sin\left(-\theta\right) = -\sin\left(\theta\right)\)[/tex]:
[tex]\[
\sin\left(-\frac{3\pi}{8}\right) = -\frac{\sqrt{2 + \sqrt{2}}}{2}
\][/tex]
Hence, the exact value of [tex]\(\sin\left(-\frac{3\pi}{8}\right)\)[/tex] is:
[tex]\[
\boxed{-0.9238795325112867}
\][/tex]
