The tangent of an angle is the slope of the radius of the terminal side of the angle drawn in standard position

Question

How does this fact relate to the domain, range, period, and increasing nature of the function?

Answer 1

All slopes are defined with the exception of the slope of a vertical line. Therefore, tan(θ) is defined for all values of θ except for those that result in vertical lines which occur at θ = π/2 and θ = 3π/2. These then are the domain restrictions for tan(θ).

However, the range, aka slope value or tan(θ) value, can be any value from negative ∞ to positive ∞.

The period is defined from the idea of the domain restrictions. Since there is no defined value of slope at θ = π/2 and θ = 3π/2, the period of f(θ) = tan(θ) is between those two restriction and 3π/2 - π/2 = π making the period = π.

Regarding the increasing value of the function, if we consider what happens between θ = 0 and θ = π/2, the slope starts at 0 and gets larger and larger, in other words, since the values increase as θ increases, the function is increasing on the interval [0, π/2). Considering between θ = π/2 and θ = π, the slope goes from an extremely high negative value to a lower negative value and then to zero at θ = π, again meaning the function is increasing on (π/2, π]. Between θ = π and θ = 3π/2, the slopes increase again from zero to extremely large values meaning the function increases as θ moves from π/ to θ = 3π/2. And, lastly, as θ moves from 3π/2 to 2π, the slopes again move from extremely large negative values towards zero, again meaning the function is increasing on that interval.

Answer 2

What is the diagram under discussion?