Sure, let's break down the solution step-by-step to find the exact value of the given expression:
[tex]\[
\sin \left[\sin^{-1} \frac{5}{13} - \cos^{-1} \left(-\frac{4}{5}\right)\right]
\][/tex]
1. Identify and assign the angles for simplicity:
Let [tex]\(\theta_1 = \sin^{-1} \frac{5}{13}\)[/tex] and let [tex]\(\theta_2 = \cos^{-1} \left(-\frac{4}{5}\right)\)[/tex].
2. Evaluate [tex]\(\theta_1\)[/tex]:
[tex]\(\theta_1\)[/tex] is the angle whose sine is [tex]\(\frac{5}{13}\)[/tex]. We understand that in this scenario:
[tex]\[
\sin(\theta_1) = \frac{5}{13}
\][/tex]
3. Evaluate [tex]\(\theta_2\)[/tex]:
[tex]\(\theta_2\)[/tex] is the angle whose cosine is [tex]\(-\frac{4}{5}\)[/tex]. Thus:
[tex]\[
\cos(\theta_2) = -\frac{4}{5}
\][/tex]
4. Calculate the expression [tex]\(\theta_1 - \theta_2\)[/tex]:
We find the numerical values as follows to understand the relationship between the angles:
[tex]\[
\theta_1 \approx 0.3948 \text{ radians}
\][/tex]
[tex]\[
\theta_2 \approx 2.4981 \text{ radians}
\][/tex]
Therefore, the difference is:
[tex]\[
\theta_1 - \theta_2 \approx 0.3948 - 2.4981 = -2.1033 \text{ radians}
\][/tex]
5. Evaluate [tex]\(\sin(\theta_1 - \theta_2)\)[/tex]:
Now, we need to find the sine of the resultant angle:
[tex]\[
\sin(\theta_1 - \theta_2) \approx \sin(-2.1033) \approx -0.8615
\][/tex]
Thus, the exact value of the expression is:
[tex]\[
\sin \left[\sin^{-1} \frac{5}{13} - \cos^{-1} \left(-\frac{4}{5}\right)\right] \approx -0.8615
\][/tex]
