Properties of Inverse Trigonometric Functions Mathematics Notes | EduRev

Additional Topics for IIT JAM Mathematics

Mathematics : Properties of Inverse Trigonometric Functions Mathematics Notes | EduRev

The document Properties of Inverse Trigonometric Functions Mathematics Notes | EduRev is a part of the Mathematics Course Additional Topics for IIT JAM Mathematics.
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The inverse trigonometric functions are also called arcus functions or anti trigonometric functions. These are the inverse functions of the trigonometric functions with suitably restricted domains. Specifically, they are the inverse functions of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle’s trigonometric ratios.

Properties of Trigonometric Inverse Functions
Here are the properties of the inverse trigonometric functions with proof.
Properties of Inverse Trigonometric Functions Mathematics Notes | EduRev

Property 1
i. sin-1 (1/x) = cosec-1x , x ≥ 1 or x ≤ -1
ii. cos-1 (1/x) = sec-1x , x ≥ 1 or x ≤ -1
iii. tan-1 (1/x) = cot-1x , x > 0
Proof : sin-1 (1/x) = cosec-1x , x ≥ 1 or x ≤ -1,
Let  sin−1 x = y
i.e. x = cosec y
Properties of Inverse Trigonometric Functions Mathematics Notes | EduRev
Hence, Properties of Inverse Trigonometric Functions Mathematics Notes | EduRev where, x ≥ 1 or x ≤ -1.

Property 2
i. sin-1(-x) = – sin-1(x),    x ∈ [-1,1]
ii. tan-1(-x) = -tan-1(x),   x ∈ R
iii. cosec-1(-x) = -cosec-1(x), |x| ≥ 1
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
i. cos-1(-x) = π – cos-1 x, x ∈ [-1,1]
ii. sec-1(-x) = π – sec-1x, |x| ≥ 1
iii. cot-1(-x) = π – cot-1x, x ∈ R
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,  cosπ − q = −cos q
cos−1x = π − y
cos−1x = π–cos−1–x
Hence, cos−1−x = π–cos−1x

Property 4
i. sin-1x + cos-1x = π/2, x ∈ [-1,1]
ii. tan-1x + cot-1x = π/2, x ∈ R
iii. cosec-1x + sec-1x = π/2, |x| ≥ 1
Proof : sin-1x + cos-1x = π/2, x ∈ [-1,1]
Properties of Inverse Trigonometric Functions Mathematics Notes | EduRev
Hence, sin-1x + cos-1x = π/2, x ∈ [-1,1]

Property 5
tan-1x + tan-1y = tan-1((x+y)/(1-xy)), xy < 1.
tan-1x – tan-1y = tan-1((x-y)/(1+xy)), xy > -1.
Proof : tan-1x + tan-1y = tan-1((x+y)/(1-xy)), xy < 1.
Let tan−1x = A
And tan−1y = B
Then, tan A = x
tan B = y
Now, tan(A+B)=(tanA+tanB)/(1−tanAtanB)
Properties of Inverse Trigonometric Functions Mathematics Notes | EduRev
Properties of Inverse Trigonometric Functions Mathematics Notes | EduRev

Property 6
i. 2tan-1x = sin-1 (2x/(1+x2)), |x| ≤ 1
ii. 2tan-1x = cos-1((1-x2)/(1+x2)), x ≥ 0
iii. 2tan-1x = tan-1(2x/(1 – x2)), -1 < x <1
Proof : 2tan-1x = sin-1 (2x/(1+x2)), |x| ≤ 1
Let tan−1x = y and x = tan y
Properties of Inverse Trigonometric Functions Mathematics Notes | EduRev
Since, sin2θ = 2tanθ/(1+tan2θ),
= 2y
= 2tan−1x which is our LHS

Hence 2 tan-1x = sin-1 (2x/(1+x2)), |x| ≤ 1

Solved Example
Q1. Prove that “sin-1(-x) = – sin-1(x),    x ∈ [-1,1]”
Ans: Let, sin−1(−x) = y
Then −x = siny
x = −siny
x = sin(−y)
sin−1x = arcsin(sin(−y))
sin−1x = y
sin−1x = −sin−1(−x)
Hence, sin−1(−x)=−sin−1x, x ∈ [-1,1]

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