What is an Inverse Function?
f y=f(x) and x=g(y) are two functions such that f (g(y)) = y and g (f(y)) = x, then f and y are said to be inverse of each other i.e.,
g = f-1
If y = f(x) then x = f-1 (y)
Inverse Trigonometric Formulas
Fig: Inverse Trigonometric Functions
The inverse trigonometric functions are the inverse functions of the trigonometric functions written as cos-1 x, sin-1 x, tan-1 x, cot-1 x, cosec-1 x, sec-1 x.
The inverse trigonometric functions are multivalued. For example, there are multiple values of ω such that z = sinω, so sin-1z is not uniquely defined unless a principal value is defined.
Such principal values are sometimes denoted with a capital letter so, for example, the principal value of the inverse sine may be variously denoted sin-1z or arcsinz.
Let’s say, if y = sin x , then x = sin-1 y, similarly for other trigonometric functions. This is one of the inverse trigonometric formulas. Now, y = sin-1 (x), y ∈ [π/2 , π/2] and x ∈ [-1,1].
Domain and Range of Inverse Trigonometric Formulas
Ques 1: Find the exact value of each expression without a calculator, in [0,2π).
Notice how each one of these examples yields two answers. This poses a problem when finding a singular inverse for each of the trig functions. Therefore, we need to restrict the domain in which the inverses can be found.
Ques 2: The value of 6 sin-11
Ans: Let A=sin−11, then sinA=1
6 sin-1 1 = 6 * π/2
6 sin-1 1 = 3π
Ques 3: Find the value of tan-1(1.1106).
Ans: Let A=tan−1(1.1106)
Then, tanA = 1.1106
A = 48°
tan48 = 1.1106
[Use calculator in degree mode]
Properties of Inverse
Here are the properties of the inverse trigonometric functions with proof.
Proof : sin-1 (1/x) = cosec-1x , x ≥ 1 or x ≤ -1,
i.e. x = cosec y
1/π = sin y
Proof: sin-1(-x) = -sin-1(x), x ∈ [-1,1]
Let, Sin-1(-x) = y
Then −x=sin y
Hence,sin−1(−x)=−sin−1 x ∈ [-1,1]
Proof : cos-1(-x) = π – cos-1 x, x ∈ [-1,1]
cos y=−x x=−cos y
Since, cos π−q=−cos q
Ques 1: Prove that “sin-1(-x) = – sin-1(x), x ∈ [-1,1]”
Ans: Let, sin−1(−x)=y
Then −x=sin y
Hence, sin−1(−x)=−sin−1 x, x ∈ [-1,1]
This concludes our discussion on the topic of trigonometric inverse functions.
Ques 2: sin-1(cos π/3)=?
Ans: sin−1 [substitute cos(π/3)=1/2]
= π/6 [substitute sin-1 (1/2) = π/6]
Ques 3: Find the value of sin (π/4+Cos-1(√2/2)).
Then, cosA= √2/2
Multiplying the numerator as well as denominator by √2 we get:
A = π/4
y = sin
y = sin (π/2)