Question

How do you find the value of $\cot \dfrac{\pi }{2}$?

Answer 1

Hint:
Here we must know that $\sin \dfrac{\pi }{2} = 1{\text{ and }}\cos \dfrac{\pi }{2} = 0$ from their graph so we can substitute their value directly in the formula of $\cot x = \dfrac{{\cos x}}{{\sin x}}$ and get the required value at the required value of $x$.

Complete step by step solution:
Here we are given that we need to find the value of $\cot \dfrac{\pi }{2}$
So we know that $\pi {\text{ radian}} = 180^\circ $ so we can say that $\dfrac{\pi }{2}{\text{radian}} = 90^\circ $
Now we know the formula of the trigonometric function $\cot x$ that it is the ratio of the $\cos x{\text{ and }}\sin x$
So we can write it as:
$\cot x = \dfrac{{\cos x}}{{\sin x}}$
Now we know that we need to find the value of $\cot \dfrac{\pi }{2}$ therefore we need to substitute the value of the variable $x$ according to our required value. As we know that $\sin \dfrac{\pi }{2} = 1{\text{ and }}\cos \dfrac{\pi }{2} = 0$ as these are the general values which needs to be remembered by the student.
We need the value of $\cot \dfrac{\pi }{2}$
So if we will put $x = \dfrac{\pi }{2}$ in the formula of the$\cot x$, we will get the value of this which is required of us.
Now we will get the term after substituting $x = \dfrac{\pi }{2}$ in the formula of $\cot x$ which is $\cot x = \dfrac{{\cos x}}{{\sin x}}$ we will get:
$\cot x = \dfrac{{\cos x}}{{\sin x}}$
$\cot \dfrac{\pi }{2} = \dfrac{{\cos \dfrac{\pi }{2}}}{{\sin \dfrac{\pi }{2}}}$$ - - - - (1)$
As we know that the value of $\sin \dfrac{\pi }{2} = 1{\text{ and }}\cos \dfrac{\pi }{2} = 0$
Hence we can substitute it in the equation (1) and we will get the desired value of the trigonometric function which is $\cot \dfrac{\pi }{2}$
$\cot \dfrac{\pi }{2} = \dfrac{{\cos \dfrac{\pi }{2}}}{{\sin \dfrac{\pi }{2}}}$$ = \dfrac{0}{1}$
Now we know that whenever we divide $0$ any term we will get the result as $0$
Hence we can say that
$\cot \dfrac{\pi }{2} = \dfrac{{\cos \dfrac{\pi }{2}}}{{\sin \dfrac{\pi }{2}}}$$ = \dfrac{0}{1}$$ = 0$
Hence we get the value of $\cot \dfrac{\pi }{2}$$ = 0$
Hence we can say that we need to know all the general values of the $\sin {\text{ and cos}}$ trigonometric functions in order to get the value of other trigonometric functions easily.

Note:
The student must know that if were told to find the value of the $\tan \dfrac{\pi }{2}$ then the procedure would be same but then we would have applied the formula of $\tan x$ which says that:
$\tan x = \dfrac{{\sin x}}{{\cos x}}$
So for all such problems basic values of $\sin {\text{ and cos}}$ are to be remembered.