Trigonometric Functions: Understanding Domain and Range

Domain and Range of Basic Trigonometric Functions (Sin, Cos, Tan)

Let's start with the most fundamental trigonometric identity:

sin ² x + cos ² x = 1

From this identity, we can deduce the following:

cos ² x = 1- sin ² x

cos x = √(1- sin ² x)

As we know, the cosine function is defined for real values, hence the value inside the root must always be non-negative. Thus,

1- sin ² x ≥ 0

sin ² x ≤ 1

sin x ∈ [-1, 1]

Thus, we have the range and domain for the sine function.

Following the same process,

1- cos ² x ≥ 0

cos ² x ≤1

cos x ∈ [-1,1]

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

• -1 ≤ sin x ≤1
• -1 ≤ cos x ≤1

Now, let's discuss the function f(x)= tan x. We know, tan x = sin x / cos x. This implies that tan x will be defined for all values except those that make cos x = 0, because a fraction with denominator 0 is undefined. We know that cos x is zero for the angles π/2, 3π/2, 5π/2 etc. hence,

Therefore, tan x is not defined for these values.

Domain and Range for Sec, Cosec and Cot Functions

We know that sec x, cosec x and cot x are reciprocals of cos x, sin x and tan x respectively. Thus,

sec x = 1/cos x

cosec x = 1/sin x

cot x = 1/tan x

Therefore, 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.