To find the exact value of [tex]\(\tan \left(\sin ^{-1}\left(-\frac{3}{5}\right)\right)\)[/tex], we need to go through a few steps to understand the problem properly.
1. Understand the Inverse Sine Function:
[tex]\(\sin^{-1}\left(-\frac{3}{5}\right)\)[/tex] represents an angle [tex]\(\theta\)[/tex] such that [tex]\(\sin(\theta) = -\frac{3}{5}\)[/tex].
2. Identify the Known Values:
Knowing [tex]\(\sin(\theta) = -\frac{3}{5}\)[/tex], we can imagine a right-angled triangle where the opposite side to the angle [tex]\(\theta\)[/tex] is [tex]\(-3\)[/tex] (since [tex]\(\sin\)[/tex] is negative) and the hypotenuse is [tex]\(5\)[/tex].
3. Use the Pythagorean Theorem:
To find the adjacent side of the triangle, we can use the Pythagorean theorem:
[tex]\[
(\text{hypotenuse})^2 = (\text{opposite})^2 + (\text{adjacent})^2
\][/tex]
Plugging in the values we have:
[tex]\[
5^2 = (-3)^2 + (\text{adjacent})^2
\][/tex]
[tex]\[
25 = 9 + (\text{adjacent})^2
\][/tex]
[tex]\[
(\text{adjacent})^2 = 25 - 9
\][/tex]
[tex]\[
(\text{adjacent})^2 = 16
\][/tex]
[tex]\[
\text{adjacent} = 4
\][/tex]
4. Consider the Sign of the Adjacent Side:
Since the original angle [tex]\(\theta\)[/tex] is in the fourth quadrant (where sine is negative and cosine is positive), the adjacent side is positive.
5. Compute the Tangent:
The tangent of an angle in a right triangle is given by the ratio of the opposite side to the adjacent side.
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Plugging in our values:
[tex]\[
\tan(\theta) = \frac{-3}{4}
\][/tex]
6. Result:
The exact value of [tex]\(\tan \left(\sin^{-1}\left(-\frac{3}{5}\right)\right)\)[/tex] is therefore:
[tex]\[
-0.15
\][/tex]
Thus, the exact value of [tex]\(\tan \left(\sin^{-1}\left(-\frac{3}{5}\right)\right)\)[/tex] is [tex]\(-0.15\)[/tex].
