Trigonometric Functions - Domain and Range

All trigonometric functions are basically the trigonometric ratios of any given angle. For example if we take the functions, f(x)=sin x, f(z) = tan z , etc, we are considering these trigonometric ratios as functions. Since they are considered to be functions, they will have some domain and range.

sin2x + cos2 x = 1

From the given identity, following things can be interpreted:

cos2x = 1- sin2 x

cos x = √(1- sin2x)

Now we know that cosine function is defined for real values therefore the value inside the root is always non-negative. Therefore,

1- sin2x ≥ 0

sin2x ≤ 1

sin x ∈ [-1, 1]

Similarly, following the same methodology,

1- cos2x ≥ 0

cos2x ≤1

cos x ∈ [-1,1]

Hence for the trigonometric functions f(x)= sin x and f(x)= cos x, the domain will consist of the entire set of real numbers, as they are defined for all the real numbers. The range of f(x) = sin x and f(x)= cos x will lie from -1 to 1, including both -1 and +1, i.e.

-1 ≤sin x ≤1

-1 ≤cos x ≤1

Now, let us discuss about the function f(x)= tan x. We know, tan x = sin x / cos x. It means that tan x will be defined for all values except the values that will make cos x = 0, because a fraction with denominator 0 is not defined. Now we know that cos x is zero for the angles  π/2, 3 π/2, 5 π/2 etc therefore,

Hence for these values tan x is not defined.

So, the domain of f(x) = tan x  will be R – \(\frac{(2n+1)π}{2}\)  and the range will be set of all real numbers, R. We know that sec x, cosec x and cot x are the reciprocal of cos x, sin x and tan x respectively. Thus,

sec x = 1/cosx

cosec x = 1/sinx

cot x = 1/tanx

Hence, these ratios will not be defined for the following:

  1. sec x will not be defined at the points where cos x is 0. Hence the domain of sec x will be R-(2n+1)π/2, where n∈I. The range of sec x will be R- (-1,1). Since, cos x lies between -1 to1, so sec x can never lie between that region.
  2. cosec x will not be defined at the points where sin x is 0. Hence the domain of cosec x will be R-nπ, where n∈I. The range of cosec x will be R- (-1,1). Since, sin x lies between -1 to1, so cosec x can never lie in the region of -1 and 1.
  3. cot x will not be defined at the points where tan x is 0. Hence the domain of cot x will be R-nπ, where n∈I. The range of cot x will be the set of all real numbers, R.