Mathematics. Inverse Trigonometric Functions Session.
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Transcript of Mathematics. Inverse Trigonometric Functions Session.
Session Objectives
2. Properties of Inverse Trigonmetric Function
1. Basic Concepts of inverse trigonometric functions
• Definition• Domain and Range
3. Conversion of one form of Inverse Trig. Fn. to other Forms
4. Identities containing inverse trigonometric functions
Basic Concepts - Definition
Inverse of a Function :
Function must be bijective
Trigonometric functions are periodic.
sin x is periodic with period equal to 2π
Not Bijective
Hence , inverse of sin x should not be valid ????
Basic Concepts - Definition
However, trigonometric functions are bijective for particular value sets in the domain.
sin-1 x is valid in these value sets
Inverse trigonometric function - principal value set
Smallest Numerical Angle
sin x is bijective in [-π/2 , π/2 ] and in [π/2 ,3π/2 ] ……. for x R
[-π/2 , π/2 ]
Basic Concepts - Definition
Inverse trigonometric functions inverse circular function.
arc sin x sin-1x ( principal value )
Inverse Trigonometric function - sin –1 x
Domain of sin –1x
( value which x can take ) is [-1,1]
Range of sin –1 x
(values which sin–1x can take ) is [- /2, /2 ]
-1 1/2
-/2
3/2
X
Y
Inverse Trigonometric function - cos–1 x
Domain of cos –1x
( value which x can take ) is [-1,1]
Range of cos –1 x
(values which cos–1x can take ) is [0, ]
X-1 1
/2
-
Y
Inverse Trigonometric function - tan–1 x
Domain of tan –1x
( value which x can take ) is (- , )
Range of tan –1 x
(values which sec–1x can take ) is (-/2, /2)
X
/2
-/2
-
Inverse Trigonometric function - sec–1 x
Domain of sec –1x ( value which x can take ) is (- -1] U [1, )
Range of sec –1 x
(values which sec–1x can take ) is [0, ] excl. x = /2
Y
X
/2
-/2
-
1-1
Inverse Trigonometric function - cot–1 x
Domain of cot –1x
( value which x can take ) is (- , )
Range of cot –1 x
(values which cot–1x can take ) is (0, )
Y
X
/2
-/2
-
Inverse Trigonometric function - Domain and Range
Function Domain Range
sin-1x [-1,1] [-/2, /2]
cos-1x [-1,1] [0, ]
tan-1x (- , ) (-/2, /2)
sec-1x (-, -1] U [1, ) [0, ] excl. /2
cosec-1x (-, -1] U [1, ) [-/2, /2] excl. 0
cot-1x (- , ) (0, )
Function Domain Range
Inverse Trigonometric function – Properties
Always remember to keep the constraint of domain and range , while solving inverse trigonometric functions.
sin( sin-1 x) = x and cos (cos-1 x) = x if x is in [-1,1]
tan( tan-1 x) = x x if x is in ( -, )
sin-1 ( sin x) = x if x is in [-/2, /2]
cos-1 ( cos x) = x if x is in [0, ]
sec-1 ( sec x) = x if x is in [0, ] excl. x = /2
Inverse Trigonometric function – Properties
cos-1 (-x) = - cos-1x if x is in [-1,1]
sin-1 (-x) = -sin-1 (x) if x is in [-1,1]
tan-1(-x) = - tan-1x if x is in ( -, )
cot-1(-x) = - cot-1x if x is in (-,)
cosec-1(-x) = - cosec-1x if x is in (-,-1] U [1,)
sec-1(-x) = - sec-1x if x is in (-,-1] U [1,)
Inverse Trigonometric function – Properties
Always remember to keep the constraint of domain and range , while solving inverse trigonometric functions.
sin-1 (-x) = -sin-1 (x) if x is in [-1,1]
Let y = sin-1(-x) ; constraint : y is in [-/2, /2]
sin y = - x x = - sin y = sin ( -y )
sin-1(-x) = sin-1 ( sin (-y))
sin-1(-x) = -y
sin-1(-x) = -sin-1 x
Class Exercise - 1
Find the principal value of
1 2sin sin
3
Solution : 1 2
Let sin sin3
2sin sin where ,
3 2 2
3sin
2
3
Class Exercise - 2
Find the principal value of sin –1 ( sin 5 )
Let y = sin-1(sin 5).Hence y is in [-/2,/2]
5 Wrong
Now , sin 5 = sin [(5/). ] = sin ( 1.59)
= - sin (2 - 1.59)
= sin ( 1.59 -2)
in [-/2, /2]
sin 5 = sin ( 5 - 2 )
sin-1(sin 5) = sin-1 ( sin ( 5 - 2 ))
= 5 - 2
Solution :
Other important properties
1 1 1 x ytan x tan y tan
1 xy
If x > 0 , y > 0 and xy < 1
1 1 1 x ytan x tan y tan
1 xy
If x > 0 , y > 0 and xy > 1
1 1 1 x ytan x tan y tan
1 xy
If x<0,y<0 and xy < 1
sin-1 x+ cos-1 x = /2 ;
if x is in [-1,1]
Class Exercise - 5
Find the value of
1 11 1tan tan
2 3
Solution :
1 11 1Let tan and tan
2 3
1 1
tan and tan and , ,2 3 2 2
1 12 3Now tan1 1
1 .2 3
tan 1
Class Exercise - 9
In triangle ABC if A = tan-12 and B = tan-1 3 , prove that C = 450
Solution : For triangle ABC , A+B+C =
tan A B tan C
tanA tanBtanC
1 tanA.tanB
tanA 2; tanB 3
3 2tanC
1 3.2
1 tanC tanC 1
Inverse Trigonometric function – Conversion
To convert one inverse function to other inverse function :
1. Assume given inverse function as some angle ( say )
2. Draw a right angled triangle satisfying the angle. Find the third un known side
3. Find the trigonometric function from the triangle in step 2. Take its inverse and we will get = desired inverse function
Conversion - Illustrative Problem
The value of cot-1 3 + cosec-1 5 is
(a) /3 (b) /2 ( c) /4 (d) none
Step 1
Assume given inverse function as some angle ( say )
Let cot-1 3 + cosec-1 5 = x + y,
Where x = cot-13 ; cot x = 3 and
y = cosec-1 5 ; cosec y = 5
Conversion - Illustrative Problem
The value of cot-1 3 + cosec-1 5 is
(a) /3 (b) /2 ( c) /4 (d) none
Step 2
Draw a right angled triangle satisfying the angle. Find the third unknown side
x1
3
10
y 5 1
2
cot x = 3 , tan x = 1/3
cosec y = 5 , tan y = 1/2
Conversion - Illustrative Problem
The value of cot-1 3 + cosec-1 5 is
(a) /3 (b) /2 ( c) /4 (d) none
Step 3
Find the trigonometric function from the triangle in step 2. Take its inverse and we will get = desired inverse function
ytanxtan1ytanxtan)yxtan(
21.
31121
31
)yxtan(
tan ( x+ y) = 1
x + y = /4
tan x = 1/3 ,tan y = 1/2
Conversion - Illustrative Problem
Prove that
sin cot-1 tan cos-1 x = x
Step 1
Assume given inverse function as some angle ( say )
Let y = sin cot-1 tan cos-1 x
And cos-1 x = , cos = x
Hence , y = sin cot-1 tan
Solution :
Conversion - Illustrative Problem
Prove that sin cot-1 tan cos-1 x = x
Step 2
Draw a right angled triangle satisfying the angle. Find the third unknown side
1
x
2x1 cos = x, tan =xx1 2
Hence , y = sin cot-1
xx1 2
y = sin cot-1 tan
Conversion - Illustrative Problem
Prove that sin cot-1 tan cos-1 x = x
Step 2
Draw a right angled triangle satisfying the angle. Find the third unknown side
1 x
2x1 and y = sin
From the adjoining triangle , sin = x
Let cot-1 = , cot =xx1 2
xx1 2
Hence y = x = R.H.S.
y = sin cot-1
xx1 2
Class Exercise - 3
Find the value of
1 1tan 2tan
3
Solution : 1 1Let 2 tan
3
1
tan2 3
2
12.2 tan 32As tan , tan
11 tan 1
2 9
3tan
4
Class Exercise - 4
Find the value of
11 5tan cos
2 3
Solution : 11 5 5
cos cos22 3 3
2
511 cos2 3tan
1 cos2 51
3
2 3 5tan
3 5
223 5
tan4
3 5tan
2
Class Exercise - 6
Prove that
1 1 1 2tan x cot x 1 tan x x 1
Solution :
1 1Let tan x and cot x 1
1tan x and cot x 1 tan
x 1
And L.H.S. of the given identity is +
1
x tan tanx 1tan as tan1 1 tan .tan1 x.
x 1
Class Exercise - 6
Prove that
1 1 1 2tan x cot x 1 tan x x 1
Solution : given identity is +
1
xx 1and tan
11 x.
x 1
2x x 1
tanx 1 1
2tan x x 1
1 2tan x x 1
Class Exercise - 7
Solve the equation
1 15 12sin sin
x x 2
Solution : 1 15 12
Let sin and sinx x
5 12sin and sin
x x
given equation is2
cos 0 cos .cos sin .sin
Class Exercise - 7
Solve the equation 1 15 12
sin sinx x 2
Solution :
cos 0 cos .cos sin .sin
2
2
5 25 xAs sin cos
x x
12 144 xand sin cos
x x
2 225 x 144 x 5 12. .
x x x x
Class Exercise - 7
Solve the equation 1 15 12
sin sinx x 2
Solution : 2 225 x 144 x 5 12
. .x x x x
2 225 x . 144 x 60 22 225 x . 144 x 60
4 2x 169x 0 x 0, 13 , x 0
Now If x 13 , , 0, hence x 13
Class Exercise - 8
If sin-1 x + sin-1 (1- x) = cos-1x,
the value of x could be
(a) 1, 0 (b) 1,1/2 (c) 0,1/2 (d) 1, -1/2
Class Exercise - 8If sin-1 x + sin-1 (1- x) = cos-1x,
the value of x could be
(a) 1, 0 (b) 1,1/2 (c) 0,1/2 (d) 1, -1/2
Solution :
1 1Let sin x and sin 1 x
sin x and sin 1 x
and given equation is + = cos-1x cos (+) = x
2
2 2
As sin x cos 1 x
and sin 1 x cos 1 1 x 2x x
Class Exercise - 8If sin-1 x + sin-1 (1- x) = cos-1x,
the value of x could be
(a) 1, 0 (b) 1,1/2 (c) 0,1/2 (d) 1, -1/2
Solution : cos (+) = x
2 2sin x , cos 1 x , sin 1 x , cos 2x x
2 21 x . 2x x x. 1 x x
2 2 21 x . 2x x 2x x
22 2 21 x . 2x x 2x x
2 2 22x x 1 x 2x x 0
Class Exercise - 8If sin-1 x + sin-1 (1- x) = cos-1x,
the value of x could be
(a) 1, 0 (b) 1,1/2 (c) 0,1/2 (d) 1, -1/2
Solution : 2 2 22x x 1 x 2x x 0
x. 2 x 1 2x 0
1x 0, ,2
2
x 2 as x sin 1
x 0,2
Class Exercise - 10
If cos-1 x + cos-1 y + cos-1z = ,
Then prove that x2+y2+z2 = 1 - 2xyz
1 1 1Let cos x A , cos y B and cos z C
Solution :
cos A x, cosB y and cosC z and given : A+B+C =
Now, L.H.S. = cos2A + cos2B +cos2C
= cos2A + 1- sin2B +cos2C
= 1+(cos2A - sin2B) +cos2C
Class Exercise - 10
If cos-1 x + cos-1 y + cos-1z = ,
Then prove that x2+y2+z2 = 1 - 2xyz
Solution : Given : A+B+C =
L.H.S. = 1+(cos2A - sin2B) +cos2C
= 1+ cos(A+B).cos(A–B) +cos2C
= 1+ cos( – C).cos(A–B) +cos2C
= 1+ cosC [– cos(A–B) +cosC ]
= 1+ cosC [– cos(A–B) – cos(A+B)]
Class Exercise - 10
If cos-1 x + cos-1 y + cos-1z = ,
Then prove that x2+y2+z2 = 1 - 2xyz
Solution : Given : A+B+C =
L.H.S. = 1+ cosC [– cos(A–B) – cos(A+B)]
= 1– cosC [ cos(A–B) + cos(A+B)]
= 1– cosC [2cos A. cos B]
= 1– 2xyz = R.H.S.