# 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.

• 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 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} \$)

#### 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} \$) 