To determine the value of [tex]\(\tan 45^\circ\)[/tex], we start by knowing the fundamental definition of the tangent function for an angle in right triangle trigonometry. The tangent of an angle [tex]\(\theta\)[/tex] is given by 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]
For a [tex]\(45^\circ\)[/tex] angle in a right triangle, it's helpful to consider an isosceles right triangle where the two legs are equal in length. Let's assume the legs are both of length 1 unit.
So, in an isosceles right triangle with both legs having a length of 1 unit:
[tex]\[
\tan 45^\circ = \frac{\text{opposite leg}}{\text{adjacent leg}} = \frac{1}{1} = 1
\][/tex]
Let's confirm we are on the same page with the trigonometric values for this key angle. To convert [tex]\(45^\circ\)[/tex] to radians (if you're more comfortable working in radians), recall that:
[tex]\[
45^\circ = \frac{\pi}{4} \text{ radians}
\][/tex]
Using the unit circle, the coordinates at [tex]\(\frac{\pi}{4}\)[/tex] radians or [tex]\(45^\circ\)[/tex] are [tex]\((\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})\)[/tex]. Therefore,
[tex]\[
\tan 45^\circ = \frac{\sin 45^\circ}{\cos 45^\circ} = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = 1
\][/tex]
Finally, considering the numerical result:
[tex]\[
\tan 45^\circ \approx 0.9999999999999999
\][/tex]
which is very close to 1. Hence, the most precise answer matching the value of [tex]\(\tan 45^\circ\)[/tex] is:
[tex]\[
\boxed{1}
\][/tex]
