To determine the value of [tex]\(\tan 45^\circ\)[/tex], let's consider a few important trigonometric concepts.
The tangent function for any given angle [tex]\(\theta\)[/tex] in a right-angled triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side:
[tex]\[
\tan \theta = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Now, for the angle [tex]\(45^\circ\)[/tex], we know that it corresponds to a special case in trigonometry where the sides of a 45°-45°-90° triangle (an isosceles right-angled triangle) are in a known ratio. Specifically, for such a triangle, both the legs (opposite and adjacent sides for [tex]\(45^\circ\)[/tex]) are of equal length.
Let's assume each leg has a length of 1 unit.
Thus, the formula for [tex]\(\tan 45^\circ\)[/tex] becomes:
[tex]\[
\tan 45^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{1} = 1
\][/tex]
So, the value of [tex]\(\tan 45^\circ\)[/tex] is approximately 1. However, due to the precision limits of floating-point arithmetic, the computed result came out as 0.9999999999999999, which is extremely close to 1.
Given the provided options:
A. [tex]\(\sqrt{2}\)[/tex]
B. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
C. [tex]\(\frac{1}{2}\)[/tex]
D. 1
The closest and most accurate option is D. Therefore, the value of [tex]\(\tan 45^\circ\)[/tex] is indeed:
[tex]\[
\boxed{1}
\][/tex]
