Why is there no corresponding point on tan^-1(x) for the (135,-1) on the graph tan(x).
Thanks!
The function tan(x) is periodic, which means it repeats itself after a certain interval. In this case, the tangent function has a period of π, which means that every π units on the x-axis, the function will repeat itself.
Now, let's consider the point (135, -1) on the graph of tan(x). If we take the inverse tangent of -1, denoted as tan^(-1)(-1) or arctan(-1), it will give us the angle whose tangent is -1. In other words, we are looking for the value of x such that tan(x) = -1.
In the first quadrant, the tangent function is positive, so we won't find -1 there. However, in the second and fourth quadrant, where the tangent function is negative, we can find angles whose tangent is -1.
The principal value of arctan(-1) lies between -π/2 and π/2 in the second and fourth quadrants. So, there is a corresponding angle for tan^(-1)(-1), but it doesn't lie on the part of the graph of tan(x) you are considering, which is the interval (0, π).
To find the corresponding angle for tan^(-1)(-1) in the interval (0, π), you would need to add or subtract the period, π, to the principal value. However, in this specific case, since the interval (0, π) does not intersect with the angles whose tangent is -1, there is no corresponding point on the graph of tan(x) for the point (135, -1).