Use a calculator to find the values of the inverse trigonometric functions. Round to the nearest degree.
[tex]
\begin{array}{l}
sin ^{-1}\left(\frac{2}{3}\right)=\square^{\circ} \\
tan ^{-1}(4)=\square^{\circ} \\
\cos ^{-1}(0.1)=\square^{\circ}
\end{array}
[/tex]
Answer (1)
Evaluate sin − 1 ( 3 2 ) and round to the nearest degree: 4 2 ∘ .
Evaluate tan − 1 ( 4 ) and round to the nearest degree: 7 6 ∘ .
Evaluate cos − 1 ( 0.1 ) and round to the nearest degree: 8 4 ∘ .
The final answers are: 4 2 ∘ , 7 6 ∘ , 8 4 ∘ .
Explanation
Problem Analysis We are asked to find the values of the inverse trigonometric functions sin − 1 ( 3 2 ) , tan − 1 ( 4 ) , and cos − 1 ( 0.1 ) , rounded to the nearest degree. We will use a calculator to evaluate each of these expressions.
Calculating sin − 1 ( 3 2 ) First, we evaluate sin − 1 ( 3 2 ) . Using a calculator, we find that sin − 1 ( 3 2 ) ≈ 41.8 1 ∘ . Rounding to the nearest degree, we get 4 2 ∘ .
Calculating tan − 1 ( 4 ) Next, we evaluate tan − 1 ( 4 ) . Using a calculator, we find that tan − 1 ( 4 ) ≈ 75.9 6 ∘ . Rounding to the nearest degree, we get 7 6 ∘ .
Calculating cos − 1 ( 0.1 ) Finally, we evaluate cos − 1 ( 0.1 ) . Using a calculator, we find that cos − 1 ( 0.1 ) ≈ 84.2 6 ∘ . Rounding to the nearest degree, we get 8 4 ∘ .
Final Answer Therefore, the values of the inverse trigonometric functions, rounded to the nearest degree, are: sin − 1 ( 3 2 ) = 4 2 ∘ tan − 1 ( 4 ) = 7 6 ∘ cos − 1 ( 0.1 ) = 8 4 ∘
Examples
Inverse trigonometric functions are used in various fields such as physics, engineering, and navigation. For example, if you know the ratio of the sides of a right triangle, you can use the inverse trigonometric functions to find the angles of the triangle. In navigation, if you know the distance and the height of an object, you can use the arctangent function to find the angle of elevation.
