# Domain and range of trigonometric functions

From the definition of sine and cosine functions, we observe that they are defined for all real numbers. Further, we observe that for each real number x,

– 1 ≤ sin x ≤ 1 and – 1 ≤ cos x ≤ 1

Thus, domain of y = sin x and y = cos x is the set of all real numbers and range is the interval [–1, 1], i.e., – 1 ≤ y ≤ 1

Since cosec x = 1 / sin x , the domain of y = cosec x is the set { x : x ∈ R and x ≠ n π, n ∈ Z} and range is the set {y : y ∈ R, y ≥ 1 or y ≤ – 1}. Similarly, the domain of y = sec x is the set {x : x ∈ R and x ≠ (2n + 1) π /2 , n ∈ Z} and range is the set {y : y ∈ R, y ≤ – 1or y ≥ 1}. The domain of y = tan x is the set {x : x ∈ R and x ≠ (2n + 1) π 2 , n ∈ Z} and range is the set of all real numbers. The domain of y = cot x is the set {x : x ∈ R and x ≠ n π, n ∈ Z} and the range is the set of all real numbers

We further observe that in the first quadrant, as x increases from 0 to π/ 2 , sin x increases from 0 to 1, as x increases from π /2 to π, sin x decreases from 1 to 0. In the third quadrant, as x increases from π to 3π/ 2 , sin x decreases from 0 to –1and finally, in the fourth quadrant, sin x increases from –1 to 0 as x increases from 3π/ 2 to 2π. Similarly, we can discuss the behaviour of other trigonometric functions. In fact, we have the following table:

Remark In the above table, the statement tan x increases from 0 to ∞ (infinity) for 0 < x < π /2 simply means that tan x increases as x increases for 0 < x < π /2 and

assumes arbitraily large positive values as x approaches to π /2 . Similarly, to say that cosec x decreases from –1 to – ∞ (minus infinity) in the fourth quadrant means that cosec x decreases for x ∈ ( 3π /2 , 2π) and assumes arbitrarily large negative values as x approaches to 2π. The symbols ∞ and – ∞ simply specify certain types of behaviour of functions and variables

We have already seen that values of sin x and cos x repeats after an interval of 2π. Hence, values of cosec x and sec x will also repeat after an interval of 2π. We

Example If cos x = – 3 /5 , x lies in the third quadrant, find the values of other five trigonometric functions

Solution Since cos x = -3/5, we have sec x = -5 /3

sin2 x + cos2 x = 1, i.e., sin2 x = 1 – cos2 x

or sin2 x = 1 – 9 /25 = 16 /25

Hence sin x = ±4/5

Since x lies in third quadrant, sin x is negative. Therefore

sinx=-4/5

which also gives

cosec x = –5/4

Further, we have

tan x =sinx/cosx=4/3 and cotx=cosx/sinx =3/4