Hi, Why tangent of 90 degrees or Pi/2 is undefined?

Question

Please, explain it is thoroughly.

Thanks

Answer 1

because tan (pi/2) = sin(pi/2)/cos(pi/2) = 1/0 ==> undefined

Answer 2

First of all division by zero is not defined and since the adjacent side here becomes 0, a value can not be found by direct substitution. However, we can sometimes find an answer indirectly by the method of taking the limit i.e., when the function always tends to approach a particular value as the dependent variable approaches the value for which the function is not defined. For ex.
f(x)=(x^2-4)/(x-2) is not defined at x=2. But
if u cnsider approaching 2, approaching, but not reaching it, we may factorise the numerator and cancel(x-2) since (x-2#0).
(x^2-4)/(x-2)=(x+2)(x-2)/(x-2)=(x+2). Thus, f(2)=2+2=4

These are called removable discontinuity. But I have proved below that such a limit canNOT be taken in case of tan(pi/2).

I can not draw a diagram but please try to imagine the figure.

Let AOA' and BOB' be the co-ordinate axes with O as the Origin.Let OP be the revolving line making an angle x with the +ve x-axis. Let M be a point on OP and MN be the perpendicular to the x-axis.
As x aproaches pi/2, On tends to vanish i.e.,it decreases in magnitude and approaches 0. Please note that when ON tends to vanish, it always remains positive. However, the length of the perpendicular MN will increase. Thus tan(pi/2) seems to be positive infinty.

Let us suppose the revolving line OP crosses OB(y axis) and enters the second Quadrant. The length of MN will of course be Positive. But ON, however small it may be will become negative.
Hence, if you look it this way, their ratio i.e., tan(pi/2) seems to be negative infinity. Hence, we are not able to obtain a consistent value for tan(pi/2) and thus, it is undefined.

tan(pi/2) is a non-removable dicontinuity.

Answer 3

From the trig identity: tan(x) = sin(x)/cos(x), it is apparent from cos(90 deg)=0, and division by zero being undefined.

Consider also the definition of tangent from right (90 deg)triangles:
tangent=length opposite side/length adjacent side (with respect to the angle). In a right triangle there cannot be two right angles; such a triangle is undefined. Hence, opposite and adjacent sides in such a triangle are also undefined.

Answer 4

Tan 90 is a perfectly vertical line standing on end with you at the bottom looking at the other end of it.

Whereas Pi/2 is half of a never ending level of precision, which would never come out even either.

One is UNDEFINED, the other is IRRATIONAL.

Answer 5

tan x = sin x / cos x.

Anywhere cos x = 0, such as at x=pi/2, tan x is undefined because dividing by zero is an undefined operation.

Answer 6

tan x=oposite side /adjacent side
as xapproaches infinity the adjacentside tend to 0
anything divided byzero is indeterminate
so tan 90 or pi/2 is indeterminate