To solve the expression [tex]\(\tan (\arctan (0.7))\)[/tex] and round the result to the nearest tenth, follow these steps:
1. Understand the Function [tex]\(\arctan\)[/tex]:
- The function [tex]\(\arctan\)[/tex], also known as the inverse tangent, is used to find the angle whose tangent value is the given number.
- In this case, [tex]\(\arctan(0.7)\)[/tex] finds the angle [tex]\( \theta \)[/tex] such that [tex]\(\tan(\theta) = 0.7\)[/tex].
2. Calculate [tex]\(\arctan(0.7)\)[/tex]:
- The value of [tex]\(\arctan(0.7)\)[/tex] is approximately 0.6107 radians.
3. Understand the Function [tex]\(\tan\)[/tex]:
- The tangent function, [tex]\(\tan\)[/tex], gives the ratio of the opposite side to the adjacent side for a given angle in a right triangle.
- Here, we are looking for [tex]\(\tan(\theta)\)[/tex] where [tex]\(\theta = \arctan(0.7)\)[/tex].
4. Calculate [tex]\(\tan(\arctan(0.7))\)[/tex]:
- Since [tex]\(\theta\)[/tex] is the angle such that [tex]\(\tan(\theta) = 0.7\)[/tex], applying the tangent to this angle returns the value back to 0.7.
5. Round the Result to the Nearest Tenth:
- The value obtained before rounding is 0.7.
- When rounded to the nearest tenth, the result is still 0.7.
In conclusion, [tex]\(\tan (\arctan (0.7))\)[/tex] evaluated and rounded to the nearest tenth is:
[tex]\[
\boxed{0.7}
\][/tex]
