To find the values of the inverse trigonometric functions, we can use the definitions of the inverse sine ([tex]\(\sin^{-1}\)[/tex]), inverse tangent ([tex]\(\tan^{-1}\)[/tex]), and inverse cosine ([tex]\(\cos^{-1}\)[/tex]). We also need to convert the results from radians to degrees and then round them to the nearest degree.
1. Calculate [tex]\(\sin^{-1}\left(\frac{2}{3}\right)\)[/tex] in degrees:
- First, find [tex]\(\sin^{-1}\left(\frac{2}{3}\right)\)[/tex] using a calculator.
- Convert the result from radians to degrees.
- Round the result to the nearest degree.
- The result of [tex]\(\sin^{-1}\left(\frac{2}{3}\right)\)[/tex] is approximately 42 degrees.
2. Calculate [tex]\(\tan^{-1}(4)\)[/tex] in degrees:
- First, find [tex]\(\tan^{-1}(4)\)[/tex] using a calculator.
- Convert the result from radians to degrees.
- Round the result to the nearest degree.
- The result of [tex]\(\tan^{-1}(4)\)[/tex] is approximately 76 degrees.
3. Calculate [tex]\(\cos^{-1}(0.1)\)[/tex] in degrees:
- First, find [tex]\(\cos^{-1}(0.1)\)[/tex] using a calculator.
- Convert the result from radians to degrees.
- Round the result to the nearest degree.
- The result of [tex]\(\cos^{-1}(0.1)\)[/tex] is approximately 84 degrees.
So, the final rounded values for the inverse trigonometric functions are:
[tex]\[
\sin^{-1}\left(\frac{2}{3}\right) \approx 42^{\circ}, \quad \tan^{-1}(4) \approx 76^{\circ}, \quad \cos^{-1}(0.1) \approx 84^{\circ}.
\][/tex]
