388. find a way of describing all the intersections of the line y = 1.5399 and the graph of y = tan x. for…

Explanation:

Step1: Recall the properties of the tangent - function

The tangent function (y = \tan x) has a period of (\pi) and its domain is (x\neq\frac{\pi}{2}+k\pi,k\in\mathbb{Z}), and its range is ((-\infty,\infty)). The equation (\tan x = m) has solutions for all real - numbers (m) since the range of (y = \tan x) is ((-\infty,\infty)). To find the intersections of (y = 1.5399) and (y=\tan x), we set (\tan x=1.5399).

Step2: Use the inverse - tangent function

If (\tan x = 1.5399), then (x=\arctan(1.5399)+k\pi), where (k\in\mathbb{Z}). The function (y = \arctan t) gives the principal value of the inverse - tangent, which is in the interval ((-\frac{\pi}{2},\frac{\pi}{2})). And because the period of (y = \tan x) is (\pi), all solutions of (\tan x = 1.5399) are given by (x=\arctan(1.5399)+k\pi,k\in\mathbb{Z}).

Answer:

The equation (m = \tan x) is solvable for all real numbers (m). The intersections of (y = 1.5399) and (y=\tan x) are given by (x=\arctan(1.5399)+k\pi), where (k\in\mathbb{Z}).