Inverse Trigonometric Functions
If sin y = x, then y = sin–1x (We read it as sine inverse x)
Here, sin–1x is an inverse trigonometric function. Similarly, the other inverse trigonometric
functions are as follows:
o If cos y = x, then y = cos–1x
o If tan y = x, then y = tan–1x
o If cot y = x, then y = cot–1x
o If sec y = x, then y = sec–1x
o If cosec y = x, then y = cosec–1x
The domains and ranges (principle value branches) of inverse trigonometric functions can
be shown in a table as follows:
Range (Principle value
Function Domain
branches)
y = sin–1x [–1, 1]
y = cos–1x [–1, 1] [0, π]
y = tan–1x R
y = cot–1x R (0, π)
y = sec–1x R – (–1, 1) [0, π]
y = cosec–1x R – (–1, 1)
Note that y = tan–1x does not mean that y = (tan x)–1. This argument also holds true for the
other inverse trigonometric functions.
The principal value of an inverse tri gonometric function can be defined as
the value of inverse trigonometric functions, which lies in the range of principal branch.
Example 1: What is the principal value of ?
Solution: Let and sin–1(1) = z
and sin z = 1 = sin
We know that the ranges of principal value branch of tan–1 and sin–
1 are respectively. Also,
Therefore, principal values of
Graphs of the six inverse trigonometric functions can be drawn as follows:
The relation sin y = x ⇒ y = sin–1x gives sin (sin–1x) = x, where x ∈ [–1, 1]; and sin–1(sin x) = x,
where x ∈
This property can be similarly stated for the other inverse trigonometric functions as
follows:
o cos (cos–1x) = x, x ∈ [–1, 1] and cos–1(cos x) = x, x ∈ [0, π]
o tan (tan–1x) = x, x ∈ R and tan–1(tan x) = x, x ∈
o cosec (cosec–1x) = x, x ∈ R – (–1, 1) and cosec–1(cosec x) = x, x ∈ – {0}
o sec (sec–1x) = x, x ∈ R – (–1, 1) and sec–1(sec x) = x, x ∈ [0, π] –
o cot (cot–1x) = x, x ∈ R and cot–1(cot x)= x, x ∈ (0, π)
For suitable values of domains;
sin–1 = cosec–1 x, x ∈ R – (–1, 1)
cos–1 = sec–1 x, x ∈ R – (–1, 1)
cosec–1 = sin–1 x, x ∈ [–1, 1]
sec–1 = cos x, x ∈ [–1, 1]
Note: While solving problems, we generally use the
formulas when the conditions for x (i.e., x > 0
or x < 0) are not given
For suitable values of domains;
o sin–1 (–x) = –sin–1x, x ∈ [–1, 1]
o cos–1 (–x) = π – cos–1x, x ∈ [–1, 1]
o tan–1 (–x) = –tan–1x, x ∈R
o cosec–1 (–x) = –cosec–1x, |x| ≥ 1
o sec–1(–x) = π– sec–1x, |x| ≥ 1
o cot–1(–x) = π– cot–1x, x ∈ R
For suitable values of domains;
o sin–1x + cos–1x = , x ∈ [–1, 1]
o tan–1x + cot–1x = , x ∈ R
o sec–1x + cosec–1x = , |x| ≥ 1
For suitable values of domains;
Note: While solving problems, we generally use the
formula when the condition for xy is not given.
For x ∈ [–1, 1], 2tan–1x
For x ∈ (–1, 1), 2tan–1x
For x ³ 0, 2 tan–1x
Example: 2 For x, y ∈ [–1, 1], show that: sin–1x + sin–1y = sin–1
Solution:We know that sin–1x and sin–1y can be defined only for x, y ∈[–1, 1]
Let sin–1x = a and sin–1y = b
⇒ x = sin a and y = sin b
Also, cos a = and cos b =
We know that, sin (a + b) = sin a cos b + cos a sin b
⇒ a + b = sin–1
⇒ sin–1x + sin–1y = sin–1
Example: 3 If then find sec x.
Solution:
We have
Example: 4
Show that: 3tan–1x = tan–1
Solution: We know that,
3tan–1x = tan–1x + 2tan–1x
