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 (2)
Calculate sin − 1 ( 3 2 ) and round to the nearest degree: 4 2 ∘ .
Calculate tan − 1 ( 4 ) and round to the nearest degree: 7 6 ∘ .
Calculate cos − 1 ( 0.1 ) and round to the nearest degree: 8 4 ∘ .
The final answers are: sin − 1 ( 3 2 ) = 4 2 ∘ , tan − 1 ( 4 ) = 7 6 ∘ , and cos − 1 ( 0.1 ) = 8 4 ∘ .
Explanation
Understanding the Problem We are asked to find the values of three inverse trigonometric functions in degrees, rounded to the nearest degree. The inverse trigonometric functions are: sin − 1 ( 3 2 ) , tan − 1 ( 4 ) , and cos − 1 ( 0.1 ) .
Calculating sin − 1 ( 3 2 ) To find the value of sin − 1 ( 3 2 ) in degrees, we use a calculator. The result is approximately 41.8 1 ∘ . Rounding to the nearest degree, we get 4 2 ∘ .
Calculating tan − 1 ( 4 ) To find the value of tan − 1 ( 4 ) in degrees, we use a calculator. The result is approximately 75.9 6 ∘ . Rounding to the nearest degree, we get 7 6 ∘ .
Calculating cos − 1 ( 0.1 ) To find the value of cos − 1 ( 0.1 ) in degrees, we use a calculator. The result is approximately 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 ∘ , and cos − 1 ( 0.1 ) = 8 4 ∘ .
Examples
Inverse trigonometric functions are used in various fields such as physics, engineering, and navigation. For example, in physics, they are used to calculate angles of incidence and refraction of light. In engineering, they are used to design structures and calculate forces. In navigation, they are used to determine the direction and position of a vessel or aircraft. Understanding inverse trigonometric functions helps in solving real-world problems involving angles and triangles.
The values of the inverse trigonometric functions are as follows: sin − 1 ( 3 2 ) ≈ 4 2 ∘ , tan − 1 ( 4 ) ≈ 7 6 ∘ , and cos − 1 ( 0.1 ) ≈ 8 4 ∘ .
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