Certainly! Let's tackle the given inverse trigonometric functions step-by-step and round each result to the nearest degree.
1. Finding [tex]\(\sin^{-1}\left(\frac{2}{3}\right)\)[/tex]:
- We need to determine the angle whose sine is [tex]\(\frac{2}{3}\)[/tex].
- Using a calculator, [tex]\(\sin^{-1}\left(\frac{2}{3}\right)\)[/tex] gives an angle in radians.
- Converting this angle from radians to degrees, we get approximately 41.81 degrees.
- Rounding to the nearest degree, [tex]\(\sin^{-1}\left(\frac{2}{3}\right) \approx 42^{\circ}\)[/tex].
2. Finding [tex]\(\tan^{-1}(4)\)[/tex]:
- We need to determine the angle whose tangent is 4.
- Using a calculator, [tex]\(\tan^{-1}(4)\)[/tex] gives an angle in radians.
- Converting this angle from radians to degrees, we get approximately 75.96 degrees.
- Rounding to the nearest degree, [tex]\(\tan^{-1}(4) \approx 76^{\circ}\)[/tex].
3. Finding [tex]\(\cos^{-1}(0.1)\)[/tex]:
- We need to determine the angle whose cosine is 0.1.
- Using a calculator, [tex]\(\cos^{-1}(0.1)\)[/tex] gives an angle in radians.
- Converting this angle from radians to degrees, we get approximately 84.26 degrees.
- Rounding to the nearest degree, [tex]\(\cos^{-1}(0.1) \approx 84^{\circ}\)[/tex].
Thus, the rounded values of the inverse trigonometric functions to the nearest degree are:
[tex]\[
\begin{array}{l}
\sin ^{-1}\left(\frac{2}{3}\right)=42^{\circ} \\
\tan ^{-1}(4)=76^{\circ} \\
\cos ^{-1}(0.1)=84^{\circ}
\end{array}
\][/tex]
