# Tips and Tricks and Shortcuts on Inverse

## Tips and Tricks for Inverse

Here, In this Page Tips and Tricks for Inverse is given. 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 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($\frac{(1-x^{2} )}{(1+x^{2})}$) , x≥0

c) 2 tan-1x = tan -1($\frac{(2x )}{(1-x^ 2)}$ ), -1<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 invertible 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 (-$\frac{π}{6}$)

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