To find the size of the angle [tex]\( A \)[/tex] such that [tex]\(\tan A = 1.9\)[/tex] within the interval [tex]\( 0^\circ < A < 90^\circ \)[/tex], we need to determine the angle in degrees.
1. Understanding the problem: We are given [tex]\(\tan A = 1.9\)[/tex] and need to find the angle [tex]\( A \)[/tex] in degrees.
2. Inverse tangent function: To find the angle [tex]\( A \)[/tex], we use the inverse tangent function (also called the arctangent), which is denoted as [tex]\(\tan^{-1}\)[/tex] or [tex]\(\arctan\)[/tex]. This function will give us the angle whose tangent is 1.9.
3. Calculating the angle: By applying the inverse tangent function to 1.9, we get:
[tex]\[ A = \arctan(1.9) \][/tex]
4. Converting to degrees: The result from the arctangent function is given in radians. To convert this angle from radians to degrees, we multiply by [tex]\(\frac{180}{\pi}\)[/tex].
After performing these calculations, the angle [tex]\( A \)[/tex] is found to be approximately [tex]\( 62.24145939893998^\circ \)[/tex].
5. Rounding to match choices: We now round this value to the nearest option provided in the multiple choices. The closest option to [tex]\( 62.24145939893998^\circ \)[/tex] is [tex]\( 62.2^\circ \)[/tex].
Therefore, the size of the angle [tex]\( A \)[/tex] is [tex]\( \boxed{62.2^\circ} \)[/tex].
