Tips and Tricks and Shortcuts on Inverse

Tips and Tricks and Shortcuts for Inverse:-

As clear by name Inverse means the opposite in position, directions, etc.

In mathematical language, it is defined as a reciprocal quantity.

In this page, we will discuss two types of INVERSE

  • Trigonometric Inverse
  • Algebraic Inverse

Trigonometric Inverse Tips and Tricks and Shortcuts:

They are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions.

Property of trigonometry inverse functions

PROPERTY 1

a) sin-1( \frac{1}{x } \ ) = cosec -1 x,x≥1 or x≤ -1

b) cos-1( \frac{1}{x } \ ) = sec -1x, x≥1 or x≤-1

c) tan -1( \frac{1}{x } \ ) = cot -1x, x>0

PROPERTY 2

a) sin-1(-x) = -sin-1(x), x ∈ (-1,1)

b) tan-1 (-x) = tan-1 (x) , x ∈ R

c) cosec-1 (-x) = cosec-1 (x), |x| ≥ 1

PROPERTY 3

a) cos-1 (-x) =π-cos-1 x, X ∈ |-1,1|

b)  sec-1 (-x) = π -sec-1 x, |x|≥ 1

c) cot-1 (-x) =π – cot-1 x, X ∈ R

PROPERTY 4

a) sin-1 x+ cos-1x \frac{π}{2} \ , X∈ |-1,1|

b) tan-1 x + cot-1 x = \frac{π}{2} \ , X∈ R

c) cosec-1 + sec-1x = \frac{π}{2} \ , |x|≥ 1

PROPERTY 5

a) tan-1 x + tan-1 Y = tan-1 ( \frac{(x+y)}{(1-xy)} \ ), xy< 1

b)tan-1 x- tan -1 y = tan -1 ( \frac{(x-y)}{(1=xy)} \ ),xy > -1

PROPERTY 6

a)  2 tan-1x = sin-1( \frac{(2x)}{(1+x^2)} \ )), |x|≤1

b)  2 tan-1x = cos-1( \frac{(1-2 )}{(1+x^ 2)} \ )) , x≥1

c) 2 tan-1x = tan -1( \frac{(2x )}{(1-x^ 2)} \ )), |x|≤1

Algebraic Inverse Tips and Tricks and Shortcuts:

Inverse is a reverse of any quantity.

Addition is the opposite of subtraction; division is the opposite of multiplication, and so on.

For Example-

If, f is the inverse of y,

Then, the inverse of f(x)= 2x+3 can be written as,

f-1 (y)= \frac{ (y-3)}{2} \

Inverse Tips and Tricks and Shortcuts

QUESTION 1

Prove that the inverse of an inveritible odd function is also an odd function

Explanation

We know inverse of f is f-1
f(f-1(x)) = x

Now change x to-x
f(f-1 (x)) = -(-x)

Now change -x in terms of f (f-1(-x))
-f (f-1(-x))= f (f-1(-x))

which gives
f-1(x) = -f -1(-x)

And hence proved f-1 is also odd.

Question 2

Find the principal value of sin-1 (- \frac{1}{2} \ )

Options

A)  ( \frac{1}{2} \ )

B) (- \frac{1}{2} \ )

C) ( \frac{π}{2} \ )

D) (- \frac{π}{6} \ )

Correct Answer: D

Explanation

let  sin-1(- \frac{1}{2} \ ) = y,
then, sin y = (- \frac{1}{2} \ ) = sin ( \frac{π}{6} \ ) = sin (- \frac{π}{6} \ )

We know that the range of the principal value branch of sin-1 is (- \frac{π}{2} \ ), ( \frac{π}{2} \ )

and sin (- \frac{π}{6} \ ) = ( \frac{1}{2} \ )

Therefore the principal value of sin-1 (- \frac{1}{2} \ ) is (- \frac{π}{6} \ )

 

Inverse tips and tricks and shortcuts