**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**

**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:

**PRINCIPAL VALUES FOR INVERSE TRIGONOMETRY FUNCTION**

*. 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 -∞<x<∞ and -π/2<θ<π/2

*. cot-1(cotθ) = θ and cot(cot-1x) = x , provided -∞<x<∞ and 0<θ<π

*. 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<0

and cot-1x = tan-1(1/x) if x >0

and cot-1x = tan-1(1/x) +π if x<0

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 -∞<x<∞

*. 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<0 and x² +y² ⪯ 1)

*. 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 < 1

*. 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<x<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 < x < 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