The inverse trigonometric functions
play an important role in calculus for they serve to define m any integrals. The
concepts of inverse trigonometric functions is also used in science and Engineering
Before Learning inverse trigonometry please learn Trigonometry.
Before Learning inverse trigonometry please learn Trigonometry.
Basic
Trigonometric functions:
We have already studied
about six trigonometric functions, details of all six trigonometric functions is given as:
Domain range and period of trigonometric functions:
Note:
i. All trigonometric functions are
periodic functions.
ii.
All trigonometric functions are
Continuous in its domain.
iii.
Sin–1(x)
should not be confused with (sin x)–1.
In fact (sin x)–1 =
and similarly for other trigonometric
functions.
iv.
The value of an inverse
trigonometric functions which lies in the range of principal branch is called
the principal value of that inverse trigonometric functions.
v.
If y = f
(x) is an invertible function, then x
= f-1(y). Thus, the graph of sin–1
function can be obtained from the graph of original function by interchanging x and y axes, i.e., if (a, b)
is a point on the graph of sine
function, then (b, a) becomes the
corresponding point on the graph of inverse of sine function.
Thus, the graph of the function y = sin–1(x) can be obtained from the graph of y = sin x by interchanging x
and y axes.
Graph of Inverse trigonometric functions:
Graph of inverse trigonometric functions are obtained by making then ONE-ONE and ONTO functions by fixing their domains, because inverse of a function does not exist if functions is not ONE-ONE and ONTO functions.
If you know what is ONE-ONE and ONTO functions then you can proceed to next otherwise it's better to go and See Graph of all Trigonometric Function
If you know what is ONE-ONE and ONTO functions then you can proceed to next otherwise it's better to go and See Graph of all Trigonometric Function
Graph of inverse trigonometric functions
Graph
of y = Cos–1(x):-
Graph of inverse trigonometric functions
Graph of inverse trigonometric functions
Graph of inverse trigonometric functions
inverse trigonometric functions examples :
i.
Find the principal value of sin-1{1 }, cos-1(0
), cosec-1(2).
ii. Find
the value of tan-1(1) + cos-1(1 ) + sin-1 (1/2 ).
inverse trigonometric functions formulas
Properties
of inverse functions:-
1.
Relation
between inverse functions:
i.
Sin-1
x = cosec-1 (1/x); x É› (-∞
-1] U [1 ∞ )
ii. Cos-1 x = sec-1 (1/x) ; x É› (-∞
-1] U [1 ∞ )
iii. tan-1 x = cot-1 (1/x) ; x > 0
2.
Negative
values :
i.
Sin-1(
- x ) = – Sin-1( x ), x É› [-1
1]
ii. tan-1( - x ) = – tan-1(
x ), x É› R
iii. Cosec-1( - x ) = – cosec-1(
x ), x É› [1 ∞)
iv. Cos-1(
- x ) = Ï€ – Cos-1( x ) ; x É›
[-1 1]
v.
sec-1( - x ) = Ï€ – sec-1(
x ) ; x É› [1 ∞)
vi. cot-1(
- x ) = Ï€ – cot-1( x ) ; x É›
R.
3.
Inverse
formulas : -
i.
Sin-1(
x ) + Cos-1( x ) = π/2
, x É› [-1
1]
ii. tan-1( x ) + cot-1(
x ) = π/2, x ɛ R
iii. sec-1( x ) + cosec-1(
x ) = Ï€/2, x É› [1 ∞]
iv. tan-1(
x ) + tan-1 (y ) = tan-1[ (x+y)/(1 - xy) ] ; xy < 1, if xy > 1 the π + tan-1[(x+y)/(1 - xy)]
inverse trigonometric functions formulas are easy to remember, all formulas are derived from trigonometric formulas, just plug some trigonometric functions in places of x and y then you will realize how easy to remember.
inverse trigonometric functions examples :
1. Find the value of tan-12 + tan-13.
2. Find the value of tan-1(1/2) + tan-1
(1/5) + tan-1(1/8).
3. Find
the value of Sin-1(2/3) + Sin-1(8/17).
4. Find
the value of x in Sin-1(1 – x)
– 2Sin-1x = Ï€/2 .
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