Domain Of A Trigonometric Function

All trigonometric functions are basically the trigonometric ratios of any given angle. For case if we accept the functions, f(ten)=sin 10, f(z) = tan z, etc, we are considering these trigonometric ratios equally functions. Since they are considered to be functions, they will have some domain and range. In the upcoming word, we shall figure out the domain and range of trigonometric functions.

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

To brainstorm with, permit the states consider the simplest trigonometric identity:

sinⁱⁱx + cosⁱⁱx = one

From the given identity, the following things can be interpreted:

cosⁱⁱx = 1- sin^twox

cos ten = √(one- sin^twox)

Now we know that cosine office is defined for existent values therefore the value inside the root is ever non-negative. Therefore,

1- sin²10 ≥ 0

sin²ten ≤ ane

sin x ∈ [-1, i]

Hence, we got the range and domain for sine role.

Similarly, post-obit the same methodology,

1- cosⁱⁱx ≥ 0

cos²10 ≤1

cos x ∈ [-i,1]

Hence, for the trigonometric functions f(x)= sin ten and f(ten)= cos 10, the domain will consist of the unabridged set of real numbers, as they are defined for all the real numbers. The range of f(x) = sin ten and f(ten)= cos x volition lie from -1 to ane, including both -1 and +1, i.e.

-one ≤ sin ten ≤1
-1 ≤ cos 10 ≤i

At present, permit usa discuss the function f(ten)= tan x. We know, tan x = sin x / cos x. Information technology means that tan 10 will exist defined for all values except the values that will make cos ten = 0, considering a fraction with denominator 0 is not defined. At present, we know that cos ten is zero for the angles π/2, three π/2, five π/ii etc. therefore,

Trigonometric Functions

Hence, for these values, tan ten is not defined.

So, the domain of f(x) = tan 10 will be R –

\(\begin{array}{l}\frac{(2n+1)π}{2}\stop{array} \)

and the range will be prepare of all real numbers, R.

Domain and Range for Sec, Cosec and Cot Functions

We know that sec x, cosec 10 and cot 10 are the reciprocal of cos ten, sin x and tan x respectively. Thus,

sec x = i/cos 10

cosec x = one/sin ten

cot ten = ane/tan x

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

sec x will non exist defined at the points where cos x is 0. Hence, the domain of sec 10 will be R-(2n+1)π/two, where n∈I. The range of sec x will be R- (-1,1). Since, cos ten lies between -ane to1, and then sec x can never lie between that region.
cosec x will non be divers at the points where sin ten is 0. Hence, the domain of cosec x will exist R-nπ, where due north∈I. The range of cosec x will be R- (-1,1). Since, sin x lies betwixt -1 to1, and so cosec ten can never lie in the region of -one and i.
cot 10 will not be defined at the points where tan x is 0. Hence, the domain of cot x will be R-nπ, where due north∈I. The range of cot x will be the set of all real numbers, R.