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^{1}z 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^{1}z 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
Function  Domain  Range 
sin^{1}x  [1,1]  [π/2,π/2] 
cos^{1}x  [1,1]  [0,π] 
tan^{1}x  R  (π/2,π/2) 
cot^{1}x  R  (0,π) 
sec^{1}x  R(1,1)  [0,π]{π/2} 
cosec^{1}x  R(1,1)  [π/2,π/2]{0} 
Solved Examples
Ques 1: Find the exact value of each expression without a calculator, in [0,2π).
Ans:
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: 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]
tan^{−1 }1.1106=48°
Properties of Inverse
Here are the properties of the inverse trigonometric functions with proof.
Property 1
Proof : sin^{1 }(1/x) = cosec^{1}x , x ≥ 1 or x ≤ 1,
Let sin^{−1}x=y
i.e. x = cosec y
1/π = sin y
Property 2
Proof: sin^{1}(x) = sin^{1}(x), x ∈ [1,1]
Let, Sin^{1}(x) = y
Then −x=sin y
x=−sin y
x=sin(−y)
sin^{−1}=sin^{−1}(sin(−y))
sin^{−1}x=y
sin^{−1}x=−sin^{−1}(−x)
Hence,sin^{−1}(−x)=−sin^{−1} x ∈ [1,1]
Property 3
Proof : cos^{1}(x) = π – cos^{1} x, x ∈ [1,1]
Let cos^{−1}(−x)=y
cos y=−x x=−cos y
x=cos(π−y)
Since, cos π−q=−cos q
cos^{−1} x=π−y
cos^{−1} x=π–cos−1–x
Hence, cos^{−1}−x=π–cos^{−1}x
Solved Example
Ques 1: Prove that “sin^{1}(x) = – sin^{1}(x), x ∈ [1,1]”
Ans: Let, sin^{−1}(−x)=y
Then −x=sin y
x=−sin y
x=sin(−y)
sin^{−1} x=arcsin(sin(−y))
sin^{−1 }x=y
sin^{−1 }x=−sin^{−1}(−x)
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)).
Ans:
Let y=sin
Then, cosA= √2/2
Multiplying the numerator as well as denominator by √2 we get:
cosA=1/√2
A = π/4
Therefore
y = sin
y = sin (π/2)
hence, y=1.
209 videos443 docs143 tests

1. What is the definition of an inverse function? 
2. How do inverse trigonometric functions relate to trigonometric functions? 
3. What are some examples of inverse trigonometric functions? 
4. How can inverse trigonometric functions be useful in reallife applications? 
5. Are there any restrictions on the domains of inverse trigonometric functions? 
209 videos443 docs143 tests


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