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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.
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-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} )
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} )
Read also: Formulas for Inverse
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