Question

Choose the statement which is incorrect among the following.
I. $\cot x$ decreases from 0 to $-\infty$ in first quadrant and increases from 0 to $\infty$ in third quadrant.
II. $\sec x$ increases from $-\infty$ to $-1$ in second quadrant and decreases from $\infty$ to 1 in fourth quadrant.
III. $\mathrm{cosec}x$ increases from 1 to $\infty$ in second quadrant and decreases from $-1$ to $-\infty$ in fourth quadrant.

• A. I is incorrect
• B. II is incorrect
• C. III is incorrect
• D. IV is incorrect

Answer 1

Answer: (A)

From the figure it is clear that in the first quadrant, as $x$ increases from 0 to $\frac{\pi }{2},\sin x$ increases from 0 to 1 .

In the second quadrant, as $x$ increases from $\frac{\pi }{2}$ to $\pi ,\sin x$ decreases from 1 to 0 . In the third quadrant, as $x$ increases from $\pi$ to $\frac{3\pi }{2},\sin x$ decreases from 0 to $-1$ and finally, in the fourth quadrant, $\sin x$ increases from $-1$ to 0 as $x$ increases from $\frac{3\pi }{2}$ to $2\pi$. Similarly, we get the behaviour of other trigonometric functions and get the following table.

	I quadrant	II quadrant	III quadrant	IV quadrant
sin	increases from 0 to 1	decreases from 1 to 0	decreases from 0 to -1	increases from 0 to 1
cos	decreases from 1 to 0	decreases from 0 to -1	increases from $-1$ to 0	increases from 0 to 1
tan	increases from 0 to $\infty$	increases from $-\infty$ to 0	increases from 0 to $\infty$	increases from $-\infty$ to 0
cot	decreases from $\infty$ to 0	decreases from 0 to $-\infty$	decreases from $\infty$ to 0	decreases from 0 to $-\infty$
sec	increases from 1 to $\infty$	increases from $-\infty$ to $-1$	decreases from -1 to $-\infty$	decreases from $\infty$ to 1
cosec	decreases from $\infty$ to 1	increases from 1 to $\infty$	increases from $-\infty$ to $-1$	decreases from -1 to $-\infty$

Remark In the above table, the statement $\tan x$ increases from 0 to $-\infty$ (infinity) for $0<x<\frac{\pi }{2}$ simply means that $\tan x$ increases as $x$ increases for $0<x<\frac{\pi }{2}$ and assumes arbitrarily large positive values as $x$ approaches to $\frac{\pi }{2}$. Similarly, to say that $\mathrm{cosec}x$ decreases from $-1$ to $-\infty$ (minus infinity) in the fourth quadrant means that $\mathrm{cosec}x$ decreases for $x\in \left(\frac{3\pi }{2},2\pi \right)$ and assumes arbitrarily large negative values as $x$ approaches to $2\pi$. The symbols $\infty$ and $-\infty$ simply specify certain types of behaviour of functions and variables.
Thus, we see that all the given statements are true.