Complete Study Guide & Notes On INVERSE TRIGONOMETRIC FUNCTIONS A Formulae Guide By OP Gupta Gupta (Indira Award Winner) The essence of Mathematics is not to make simple things complicated, but to make complicated things simple! IMPORTANT TERMS, DEFINITIONS & RESULTS
A list of formulae for Trigonometric Trigonometric Functions (class XI) has been added for reference at the end of this formulae guide. 01. Basic Introduction: A function f : A B is said to be invertible if f is bijective (i.e., (i.e., one-one 1
and onto). onto ). The inverse of the function f function f iis denot ed by f : B A such that f ( y ) x if f ( x ) y , x A, y B . As trigonometric functions are many-one so, their inverse doesn’t exist. But they become one-one onto by restricting their domains. Therefore, inverse of trigonometric trigonometric functions are a re defined with restricted domains. In fact, in the discussion below we have used all the restrictions required so that the inverse of the concerned trigonometric functions do exist. If these restrictions are removed, the terms will represent inverse trigonometric relations and not the functions. Note that the inverse trigonometric functions are also called as Inverse Circular Functions. 02. List of Formulae and their proofs for Inverse Trigonometric Trigonometric Functions: 1 1 A. a) sin 1 x cos ec1 , x 1,1 b) cosec 1 x sin1 , x , 1 1, x x 1 1 c) cos 1 x sec 1 , x 1,1 d) sec 1 x cos 1 , x , 1 1, x x 1 1 1 1 cot x , x 0 tan x , x 0 e) tan 1 x f) cot 1 x π cot 1 1 , x 0 π tan 1 1 , x 0 x x PROOF a) Let sin 1( x ) θ then, sinθ x
cosecθ =
1 x
1 θ = cosec 1
x
1 1 1 sin x cosec x Other results can be proved in the same way!
B.
[H.P.]
a) sin 1 x sin 1 x, x 1,1
cos 1 x, x 1,1 b) cos1 x π co
c) tan 1 x ta n1 x, x R
d) cosec1 x cosec1 x , | x | 1
e) se c 1 x π sec 1 x , | x | 1
f) cot 1 x π cot 1 x, x R
PROOF b) Let cos1 x θ then, cosθ x
cos θ x cos π θ x θ π cos1 x
cos 1 x π θ
cos 1 x π cos 1 x
Similarly, we can prove other ot her results! π π C. a) sin 1 sin x x, x 2 2 π π c) tan 1 tan x x, x 2 2 e) se c 1 sec x x , 0 x π, x
[H.P.]
b) cos1 cos x x, 0 x π d) cosec1 cose c x x, π 2
π 2
π
x , x 0 2
f) cot 1 cot x x, 0 x π
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Inverse Trigonometric Functions By OP Gupta ( INDIRA AWARD Winner , Elect. & Comm. Engineering )
PROOF a) Let sin 1
θ
... i
then, sin θ x
... ii Substituting the value of x in (i) from (ii) we get, sin 1 sin θ θ (Replacing θ by x) sin 1 sin x x For other results we can proceed similarly! D.
[H.P.]
π , x 1,1 2 π b) tan 1 x cot 1 x , x R 2 π c) cosec1 x sec1 x , | x | 1 i.e., x 1 or x 1 2
a) sin 1 x cos 1 x
PROOF a) Let sin 1( x) θ then, sinθ x
cos θ x π
2
π
θ cos 1 x
2
π 2
cos 1 x θ
π
sin 1 x cos 1 x , 1 x 1
[H.P.]
2 Similarly, proceed for other results!
E.
a) sin 1 sin 1 y sin 1 x 1 y 2 y 1 x 2
b) cos1 cos 1 y cos 1 xy 1 x 2 1 y 2
1 x y tan , xy 1 1 xy x y 1 1 c) tan x tan y π tan 1 , x 0, y 0, xy 1 1 xy π tan1 x y , x 0, y 0, xy 1 1 xy x y tan 1 , xy 1 1 xy x y 1 1 d) tan x tan y π tan 1 , x 0, y 0, xy 1 1 xy π tan 1 x y , x 0, y 0, xy 1 1 xy x y z xyz 1 xy yz zx
e) tan 1 x tan 1 y tan 1 z tan 1
PROOF a) Let sin 1 x θ and sin 1 y β . Then, sinθ x and sinβ y . Now sin(θ β ) sin θcosβ cosθsinβ
sinθ 1 sin2 β 1 sin 2 θ sinβ
sin(θ +β)
1 y 2 y 1 x 2
θ β sin 1 1 y 2 y 1 x 2 sin 1 x sin 1 y sin 1 x 1 y 2 y 1 x 2
[H.P.]
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A Complete Formulae Guide
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b) cos1 cos 1 y cos 1 xy 1 x 2 1 y 2
Do yourself. Proceed as in (a). c) Let tan 1 x θ and tan 1 y β . Then, tanθ tan θ tan β x y Now tan θ β 1 tan θ tan β 1 xy
and tanβ y .
x y 1 y
θ β tan 1
x y tan 1 x tan 1 y tan 1 1 xy For x 0 , tan 1 x will be a positive angle and for y 0 , tan 1 will also be a positive angle. Therefore, LHS of (c) will be a positive angle and hence RHS should also be a positive angle. x y is positive. Case I When x 0, y 0 and xy 1 then 1 xy
y So, tan 1 will be a positive angle. 1 xy x y Hence, tan 1 x tan 1 y tan 1 1 y Case II When x 0, y 0 and y 1 then
[H.P.]
x y 1 xy
is negative.
y So, tan 1 will be a negative angle. Therefore we add π to make it positive and balanced. 1 xy x y 1 y
Hence, tan 1 x tan 1 y π tan 1
Case III When x 0, y 0 and xy 1 then
x y 1 y
[H.P.]
is positive.
x y So, tan 1 x tan 1 y will be a negative angle and tan 1 will be a positive angle. Therefore to 1 xy balance it we will be adding – π .
y 1 xy
Hence, tan 1 x tan 1 y π tan 1
[H.P.]
d) Let tan 1 x θ and tan 1 y β . Then, tanθ x and tanβ y . tan θ tan β x y Now tan θ β 1 tan θ tan β 1 xy
x y θ β tan 1 1 y
x y tan 1 x tan 1 y tan 1 1 xy
…(A)
Case I When x 0, y 0 and xy 1 then
x y 1 xy
is positive (or negative depending upon the
π π π absolute value of the angles x and y). Also if x 0, y 0 then, θ,β 0, . So θ β , i.e., 2 2 2 π π tan 1 x tan 1 y , . 2 2
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Inverse Trigonometric Functions By OP Gupta ( INDIRA AWARD Winner , Elect. & Comm. Engineering )
x y , xy 1 1 y
Hence, tan 1 x tan 1 y tan 1
Case II When x 0, y 0 and xy 1 then
x y 1 xy
[By using (A)
[H.P.]
is negative. Also if x 0, y 0 then,
π π π π θ 0, ,β , 0 i.e., θ 0, , β 0, . So θ β (0, π) i.e., tan 1 x tan 1 y (0, π) . 2 2 2 2 x y As tan 1 is a negative angle. Therefore we add π in RHS of (A) to make it positive and 1 y balanced.
x y , x 0, y 0 and xy 1 1 y
Hence, tan 1 x tan 1 y π tan 1
Case III When x 0, y 0 and xy 1 then
[H.P.]
y is positive. Also if x 0, y 0 then, 1 xy
π π π π θ , 0 ,β 0, i.e., θ , 0 , β , 0 . So θ β ( π,0) i.e., tan 1 x tan 1 y ( π,0) . 2 2 2 2 x y So, tan 1 x tan 1 y will be a negative angle and tan 1 will be a positive angle. Therefore to 1 xy balance it we will be adding – π in RHS of (A).
y , x 0, y 0 and xy 1 1 xy
Hence, tan 1 x tan 1 y π tan 1 F.
1
a) 2 tan x sin
[H.P.]
1
2 x , | x | 1 2 1 x
, x0 2 1 x 1 1 2 x c) 2 tan x tan , 1 x 1 2 1 x 1
1 1 x
2
b) 2 tan x cos
PROOF a) Let tan 1 x θ then, tanθ x . 2 tan θ 2 x As sin 2θ sin 2θ 1 tan 2 θ 1 2 2 x 2x i.e., 2θ sin1 2 tan1 x sin1 2 2 1 x 1 x
[H.P.]
Other results can also be proved in the same way! 03. Principal Value: Numerically smallest angle is known as the principal value. Finding the principal value: For finding the principal value, following algorithm can be followed– STEP1– Firstly, draw a trigonometric circle and mark the quadrant in which the angle may lie. st
nd
rd
STEP2– Select anticlockwise direction for 1 and 2 quadrants and clockwise direction for 3 and 4th quadrants. STEP3– Find the angles in the first rotation. STEP4– Select the numerically least (magnitude wise) angle among these two values. The angle thus found will be the principal value. STEP5– In case, two angles one with positive sign and the other with the negative sign qualify for the numerically least angle then, it is the convention to select the angle with positive sign as principal value.
The principal value is never numerically greater than
.
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A Complete Formulae Guide
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04. Table demonstrating domains and ranges of Inverse Trigonometric functions: Inverse Trigonometric Functions i.e., f ( x )
Domain/ Values of x
Range/ Values of f ( x )
sin 1 x
[ 1, 1]
π π 2 , 2
cos1 x
[ 1, 1]
[0, π]
cosec 1
R ( 1, 1)
sec 1 x
R ( 1, 1)
π π 2 , 2 {0} π [0, π] 2
tan 1
R
π π , 2 2
cot 1
R
(0 , π )
Discussion about the range of inverse circular functions other than their respective principal value branch We know that the domain of sine function is the set of real numbers and range is the closed 3π π π π π 3π interval [–1, 1]. If we restrict its domain to , , , , , etc. then, it 2 2 2 2 2 2 becomes bijective with the range [–1, 1]. So, we can define the inverse of sine function in each of these intervals. Hence, all the intervals of sin –1 function, except principal value π π –1 –1 branch (here except of , range of sin other for sin function) are known as the 2 2 than its principal value branch. The same discussion can be extended for other inverse circular functions. (Refer Q16 in Mathematicia Vol.1 By OP Gupta) 05. To simplify inverse trigonometrical expressions, following substitutions can be considered: Expression
Substitution
a 2 x 2 or
a2 x2
x a tanθ or x a cot θ
a 2 x 2 or
a2 x2
x a sinθ or x a cosθ
x 2 a 2 or
x 2 a2
x a secθ or x a cosec θ
ax
a x
x a cos2θ
ax
or
a 2 x2 a 2 x2 x ax x ax
or or
a x or
a2 x2
x a cos 2θ 2
2
a 2 x2 ax
x a sin θ or x a cos θ 2
2
x a x
x a tan θ or x a cot θ 2
2
x
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Inverse Trigonometric Functions By OP Gupta ( INDIRA AWARD Winner , Elect. & Comm. Engineering )
Note the followings and keep them in mind: 1
The symbol sin x is used to denote the smallest angle whether positive or negative, such 1
1
1
1
that the sine of this angle will give us x. Similarly cos x, tan x , cosec x, sec x, and 1
cot x are defined. 1
You should note that sin x can be written as arcsinx . Similarly other Inverse Trigonometric
Functions can also be written as arccosx, arctanx, arcsecx etc. 1
Also note that sin x (and similarly other Inverse Trigonometric Functions) is entirely 1
1
different from ( sin x ) . In fact, sin x is the measure of an angle in Radians whose sine is x
whereas ( sin x )
1
is
1
(which is obvious as per the laws of exponents ).
sin x
Keep in mind that these inverse trigonometric relations are true only in their domains i.e.,
they are valid only for some values of ‘x’ for which inverse trigonometric functions are well defined!
Check out NCERT Textbook Part I for the Graphs of Inverse Trigonometric Functions.
Trigonometric Formulae (Only For Reference): Relation between trigonometric ratios
a) tanθ d) cotθ
sinθ
b) tanθ
cosθ cosθ
1 cotθ
e) cosecθ
sinθ
1 sinθ
Trigonometric identities 2
cosθ
a) sin 2 A 2 sin A cos A
b) sec θ 1 tan θ 2
1
f) secθ
Multiple angle formulae involving 2A & 3A
a) sin θ cos θ 1 2
c) tan θ.cot θ 1
2
A A b) sin A 2sin cos
c) cosec θ 1 cot θ 2
2
2
Addition / subtraction formulae & some related
c) cos 2 A cos A sin A 2
results
d) cos A cos
a) sin A B sin A cos B cos A sin B
2
2
2
A 2
sin 2
2
c ) cos A B cos A B cos A sin B
f) 2 cos A 1 cos 2 A
cos2 B sin 2 A
g) cos 2 A 1 2sin A
d ) sin A B sin A B sin A sin B
h) 2 sin A 1 cos 2 A
2
2
2
2
2
2
cos2 B cos 2 A e) tan A B f) cot A B
tan A tan B
i) sin2 A
1 tan A tan B cot B cot A 1 cot B cot A
2
e) cos 2 A 2cos A 1
b) cos A B cos A cos B sin A sin B 2
A
2tan A 1 tan A 2
1 tan A 2
j) cos2 A
1 tan A 2
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A Complete Formulae Guide
Compiled By OP Gupta (M.+91-9650350480 | +91-9718240480)
b) sin C sin D 2 cos
CD CD sin 2 2
l) sin 3 A 3sin A 4sin A
c) cos C cos D 2cos
CD CD cos 2 2
m) cos 3 A 4 cos A 3cos A
d) cos C cos D 2sin
k) tan2 A
2 tan A 1 tan A 2
3
3
3tan A tan A 3
n) tan3 A
CD CD sin 2 2
e) 2sin Acos B sin A B sin A B
1 3tan A 2
f) 2 cos Asin B sin A B sin A B
Transformation of sums / differences into products
g) 2cos A cos B cos A B cos A B
& vice-versa
a) sin C sin D 2sin
CD CD cos 2 2
h) 2sin A sin B cos A B cos A B
Relations in Different Measures of Angle
Angle in Radian Measure = Angle in Degree Measure × Angle in Degree Measure = Angle in Radian Measure ×
θ (in radian measure)
π 180 180 π
l r
Also followings are of importance as well:
1 o = 60, 1 = 60
1 Right angle 90o 1o =
180
1 radian = 57 o1745 or 206265 seconds .
= 0.01745 radians Approx .
General Solutions
a) sin
sin y x n ( 1 )n y , where n Z .
b) cos x cos y x 2n y , where n Z . c) tan x tan y x n y , where n Z . Relation in Degree & Radian Measures Angles in Degree
0o
30 o
Angles in Radian
0c
6
45o
c
4
60 o
c
3
90 o
c
2
180o
c
c
In actual practice, we omit the exponent ‘ c’ and instead of writing similarly for others.
270 o
3 2 c
360 o
c
2
c
we simply write
and
For the complete discussion of the Trigonometric Functions, please refer to the FORMULAE GUIDE Of TRIGONOMETRIC FUNCTIONS of Class XI. For the values of Trigonometric Ratios at Standard Angles (i.e, 0 o , 30 o , 45o , 60o and 90 o ), check the
following page.
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Inverse Trigonometric Functions By OP Gupta ( INDIRA AWARD Winner , Elect. & Comm. Engineering ) Trigonometric Ratio of Standard Angles
Degree / Radian
0o
T – Ratios
0 0
sin
30 o
45o
60 o
90 o
π
π
π
π
6 1
4 1
3
2
2
cos tan
2
0
1
sec
2
1
3
1
3
2
3
cot
2
2
2
1
3
3
cosec
0
2
2
2 1
1
2 1
1
3
1
3
2
1
0
3
Trigonometric Ratios of Allied Angles Angles T- Ratios
π 2
π
θ
2
θ
πθ
πθ
3π 2
θ
3π 2
θ
2π θ OR θ
2π θ
sin cos tan cot
cosθ sinθ
cosθ sinθ
sinθ cosθ
sinθ cosθ
cosθ sinθ
cosθ
sinθ
sinθ
cosθ
sinθ cosθ
cotθ tanθ
cotθ tanθ
cotθ tanθ
tanθ cotθ
tanθ cotθ
cosecθ secθ
tanθ cotθ secθ
tanθ cotθ
sec cosec
cotθ tanθ cosecθ
cosecθ
cosecθ secθ
cosecθ
secθ
secθ cosecθ
secθ cosecθ
secθ cosecθ
secθ
Domain and Range of Trigonometric Functions
T- Functions
Domain
Range
sin
R
[ 1, 1]
cos x
R
[ 1, 1]
tan x
{ x R : x (2n 1) π 2, n Z}
R
cot x
{ x R : x n π, n Z}
R
cosec
{ x R : x n π, n Z}
R ( 1, 1)
{ x R : x (2n 1) π 2, n Z}
R ( 1, 1)
sec x
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