To solve for the value of [tex]\( b \)[/tex] given that [tex]\(\tan x^\circ = \frac{4}{3}\)[/tex], we need to consider a right triangle where [tex]\(\tan x^\circ\)[/tex] represents the ratio of the lengths of the opposite side to the adjacent side. In this case, the opposite side [tex]\( = 4 \)[/tex] and the adjacent side [tex]\( = 3 \)[/tex].
We can form the right triangle with these sides and now we need to find the hypotenuse ([tex]\(b\)[/tex]). The hypotenuse can be found using the Pythagorean theorem:
[tex]\[
\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2
\][/tex]
Substitute the lengths of the opposite and adjacent sides:
[tex]\[
\text{hypotenuse}^2 = 4^2 + 3^2
\][/tex]
[tex]\[
\text{hypotenuse}^2 = 16 + 9
\][/tex]
[tex]\[
\text{hypotenuse}^2 = 25
\][/tex]
To find the hypotenuse, take the square root of 25:
[tex]\[
\text{hypotenuse} = \sqrt{25}
\][/tex]
[tex]\[
\text{hypotenuse} = 5
\][/tex]
Therefore, the value of [tex]\(b\)[/tex] is [tex]\(5\)[/tex].
So, the correct answer is:
[tex]\( b = 5 \)[/tex].
