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Inverse trigonometrc function IIT JEE study material | ITF concept booster | ITF notes | ITF important formulae.

INVERSE TRIGONOMETRY FUNCTION

INVERSE FUNCTION
If f:X⟶ Y is a function which is both one – one and onto, then it’s inverse function
f^-1:Y⟶X is defined as: y = f(x) ⟺ f^-1(y)=x
Such that , ∀x∈X , ∀y∈Y
INVERSE TRIGONOMETRY FUNCTION
Let’s take a sine function, whose domain is R and range [-1,1]. We see that this function is many -one and onto, so , it’s inverse doesn’t exist, but if we restrict the domain of the sine function to the interval [-π/2 , π/2] , then the function is:
sin: [-π/2 , π/2] ⟶  [-1,1] and given by sinθ = x
Which is one-one and onto and therefore it’s inverse exist.
And the Inverse of sine function is defined as
Sin-1: [-1,1] ⟶ [-π/2 , π/2], such that sin-1x = θ
From this we can say that if x is a real numbers between -1 and 1 then θ will be in -π/2 and π/2.
Sin-1x = θ then x = sinθ , where, -π/2⪯ θ⪯  π/2 and -1 ⪯x⪯ 1
So , the least numerical value among all the values of the angle whose sine is x , is called the principal value of sin-1x. So we can give similar definition for cos-1x , tan-1x etc.

DOMAIN AND RANGE OF INVERSE TRIGONOMETRY FUNCTION

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GRAPHS OF INVERSE TRIGONOMETRY FUNCTION
*. Graphs of inverse trigonometry function can be drawn from the knowledge of the graphs of the corresponding trigonometry function.
*. In ITF graphs can be obtained by interchanging X and Y axis.
The graphs of inverse trigonometry function are given below:

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PRINCIPAL VALUES FOR INVERSE TRIGONOMETRY FUNCTION

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*. Keep in mind that if the domain of ITF is not stated then always consider principal value of given Inverse function.

DANGER
sin-1x and (sinx)^-1 is different terms , they are not equal
∴ sin-1x ≠ sinx)^-1 , similarly for other functions.

PROPERTIES OF INVERSE TRIGONOMETRY FUNCTION

1). *. sin-1(sinθ) = θ and sin(sin-1x) = x , provided -1 ⪯x⪯ 1 and -π/2⪯ θ⪯  π/2

*. cos-1(cosθ) = θ and cos(cos-1x) = x , provided -1 ⪯x⪯ 1 and  0⪯ θ⪯  π

*. tan-1(tanθ) = θ and tan(tan-1x) = x , provided -∞

*. cot-1(cotθ) = θ and cot(cot-1x) = x , provided -∞

*. sec-1(secθ) = θ and sec(sec-1x) = x
*. cosec-1(cosecθ) = θ and cosec(cosec-1x) = x

2). *. sin-1x = csc-1(1/x) or csc-1x = sin-1(1/x)
*. cos-1x = sec-1(1/x) or sec-1x = cos-1(1/x)

*. tan-1x = cot-1(1/x) if x>0
and tan-1x = cot-1(1/x) -π if x and cot-1x = tan-1(1/x) if x >0
and cot-1x = tan-1(1/x) +π if x

3). *. Sin-1x = cos-1√(1- x²) = tan-1(x/√1-x²) =

 = cot-1(√1-x²/x) = sec-1(1/√1-x²) = csc-1(1/x)
*. Cos-1x = sin-1(√1-x²) = tan-1(√1-x²/x) =
  = Cot-1(x/√1-x²) = sec-1(1/x) = csc-1(1/√1-x²)
*. Tan-1x = sin-1(x/√1+x²) = cos-1(1/√1+x²)
   = Cot-1(1/x) = sec-1(√1+x²) = csc-1(√1+x²/x)
4). *. Sin-1x + cos-1x = π/2 , where -1 ⪯x⪯ 1
*. Tan-1x + cot-1x = π/2 , where -∞
*. Sec-1x + csc-1x = π/2 , where x⪯ -1 or 1⪯x
5). *. Sin-1x +sin-1y = sin-1(x√1-y² + y√1-x²)
 If xy⪯0 or ( xy>0 and x² +y² ⪯ 1)
*. Sin-1x – sin-1y = sin-1(x√1-y² – y√1-x²)

 If 0⪯xy or ( xy
*. Cos-1x + cos-1y = cos-1(xy – √1-y².√1-x²),. If
  |x| , |y| ⪯ 1 , 0 ⪯ x+y
*. cos-1x – cos-1y = cos-1(xy + √1-y².√1-x²)
If |x| , |y| ⪯ 1, x ⪯ y
*. Tan-1x + tan-1y = tan-1(x+y/1-xy), if xy
*. Tan-1x – tan-1y = tan-1(x-y/1+xy), if xy > -1
*. 2sin-1x = sin-1(2x√1- x²), if -1/√2 ⪯ x ⪯ 1/√2
*. 2cos-1x = cos-1(2x² – 1) ,if 0 ⪯ x ⪯ 1
*. 2tan-1x = tan-1(2x/1-x²) if -1
   2tan-1x = sin-1(2x/1+x²) , if -1 ⪯x⪯ 1
   = cos-1(1-x²/1+x²), if 0⪯x
6). *. Sin-1(-x) = – sin-1x
     *. Cos-1(-x) = π – cos-1x
     *. Tan-1(-x) = – tan-1x
     *. cot-1(-x) = π – cot-1x
7). *. 3sin-1x = sin-1( 3x – 4x^3), if -1/2 ⪯ x⪯ 1/2
      *. 3cos-1x = cos-1(4x^3 – 3x) , if 1/2 ⪯ x ⪯ 1
      *. 3tan-1x = tan-1[(3x – x^3)/(1 – 3x²)] , if
          -1/√3
8). *. tan-1x + tan-1y + tan-1z
          = tan-1[(x+y+z-xyz)/(1- xy – yz – zx)]
POINTS TO BE REMEMBERED
     
*. sin-1x , cos-1x , tan-1x can also be written as arc sinx , arc cosx, arc tanx.
*. If it’s not stated then always consider principal value of the Inverse trigonometry function.
*. If tan-1x + tan-1y + tan-1z = π/2 , then xy + yz + zx = 1
*. tan-1x + tan-1y + tan-1z = π , then x+y+z = xyz
*. If sin-1x + sin-1y + sin-1z = π/2 , then
   x^2 + y^2 + y^2 + 2xyz = 1
 *. If sin-1x + sin-1y + sin-1z = π , then
  x√1- x^2 + y√ 1-y^2 + z√1-z^2 = 2xyz
*. If cos-1x + cos-1y + cos-1z = 3π
 then xy +yz +zx = 3
*. If cos-1x + cos-1y + cos-1z = π , then
       x^2 + y^2 + y^2 + 2xyz = 1
*. If sin-1x + sin-1y + sin-1z = 3π/2 , then
      xy +yz +zx = 3
*. If sin-1x + sin-1y = θ ,then cos-1x + cos-1y = π-θ
*. If cos-1x + cos-1y = θ ,then sin-1x + sin-1y= π-θ
*. If cos-1(x/a) + cos-1(y/b) = θ , then
  (x/a)^2 + (y/b)^2 – 2xycosθ/ab = sin^2θ

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