Inverse Trigonometric Functions
1.
Introduction:
1. A function which is a bijection is only invertible.
2. Trigonometric functions are not bijections in general, but they can be made bijections by restricting
the corresponding domains.
3. Inverse of a trigonometric function exist on only specified intervals.
2. Concept:
1. Domain and range of inverse trig functions
Function
Domain
Range
graph
𝜋
𝜋
−1
[−1 , 1]
𝑆𝑖𝑛 𝑥
[− , ]
2 2
𝐶𝑜𝑠 −1 𝑥
[−1 , 1]
[0, 𝜋]
𝑇𝑎𝑛−1 𝑥
(−∞, ∞)
𝜋 𝜋
(− , )
2 2
2. Basic conversion : If𝑆𝑖𝑛−1 𝑥 = 𝛼 then 𝑥 = 𝑆𝑖𝑛𝛼; If 𝐶𝑜𝑠 −1 𝑥 = 𝛼 then 𝑥 = 𝐶𝑜𝑠𝛼; and
If 𝑇𝑎𝑛−1 𝑥 = 𝛼 then 𝑥 = 𝑇𝑎𝑛𝛼
3. Compound angles To prove : 𝑆𝑖𝑛−1 𝑥 + 𝑆𝑖𝑛−1 𝑦 = 𝑆𝑖𝑛−1 (𝑥√1 − 𝑦 2 + 𝑦√1 − 𝑥 2 )
Let 𝑆𝑖𝑛−1 𝑥 = 𝛼, 𝑆𝑖𝑛−1 𝑦 = 𝛽Then 𝑥 = 𝑆𝑖𝑛𝛼, 𝑦 = 𝑆𝑖𝑛𝛽.
And ∴ 𝐶𝑜𝑠𝛼 = √1 − 𝑥 2 , 𝐶𝑜𝑠𝛽 = √1 − 𝑦 2
Now 𝑆𝑖𝑛(𝛼 + 𝛽) = 𝑆𝑖𝑛𝛼. 𝐶𝑜𝑠𝛽 + 𝐶𝑜𝑠𝛼. 𝑆𝑖𝑛𝛽 = 𝑥√1 − 𝑦 2 + 𝑦√1 − 𝑥 2
∴ (𝛼 + 𝛽) = 𝑆𝑖𝑛−1 (𝑥√1 − 𝑦 2 + 𝑦√1 − 𝑥 2 ) ⟹ 𝑆𝑖𝑛−1 𝑥 + 𝑆𝑖𝑛−1 𝑦
= 𝑆𝑖𝑛−1 (𝑥√1 − 𝑦 2 + 𝑦√1 − 𝑥 2 )
Similarly we can prove: (i) 𝐶𝑜𝑠 −1 𝑥 + 𝐶𝑜𝑠 −1 𝑦 = 𝐶𝑜𝑠 −1 (𝑥𝑦 − √1 − 𝑥 2 √1 − 𝑦 2 )
(ii) 𝐶𝑜𝑠 −1 𝑥 + 𝐶𝑜𝑠 −1 𝑦 = 𝑆𝑖𝑛−1 (√1 − 𝑥 2 . 𝑦 − 𝑥. √1 − 𝑦 2 )
(iii) 𝑆𝑖𝑛−1 𝑥 + 𝐶𝑜𝑠 −1 𝑦 = 𝐶𝑜𝑠 −1 (√1 − 𝑥 2 . 𝑦 − 𝑥. √1 − 𝑦 2 )
Prepared by R. VamanRao, PGT Mathematics, Kendriya Vidyalaya WARANGAL
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𝑥+𝑦
𝑥𝑦−1
(iv) 𝑡𝑎𝑛−1 𝑥 + 𝑡𝑎𝑛−1 𝑦 = 𝑡𝑎𝑛−1 (1−𝑥𝑦) (v)
𝐶𝑜𝑡 −1 𝑥 + 𝐶𝑜𝑡 −1 𝑦 = 𝐶𝑜𝑡 −1 ( 𝑦−𝑥 )
𝜋
To prove 𝑠𝑖𝑛−1 𝑥 + 𝑐𝑜𝑠 −1 𝑥 = 2 , 𝑠𝑖𝑛−1 𝑥 = 𝛼, then 𝑥 = 𝑠𝑖𝑛𝛼,
𝜋
𝜋
⟹ 𝑥 = 𝑐𝑜𝑠 ( − 𝛼) ⟹ 𝑐𝑜𝑠 −1 𝑥 = ( − 𝛼)
2
2
𝜋
𝜋
−1
−1
−1
𝑐𝑜𝑠 𝑥 + 𝛼 =
⟹ 𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥 =
2
2
𝜋
𝜋
Similarly we can prove (i) 𝑡𝑎𝑛−1 𝑥 + 𝑐𝑜𝑡 −1 𝑥 = , (ii) 𝑠𝑒𝑐 −1 𝑥 + 𝑐𝑜𝑠𝑒𝑐 −1 𝑥 = .
4. Complementary angles
2
5. Multiple angles
2
To prove 2𝑠𝑖𝑛−1 𝑥 = 𝑠𝑖𝑛−1 (2𝑥√1 − 𝑥 2 )
let𝑠𝑖𝑛−1 𝑥 = 𝛼 ⇒ 𝑥 = 𝑠𝑖𝑛𝛼, 𝑎𝑛𝑑 ∴ cos 𝛼 = √1 − 𝑥 2
𝑠𝑖𝑛2𝛼 = 2𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛼 ⟹ 𝑠𝑖𝑛2𝛼 = 2𝑥√1 − 𝑥 2 ⟹ 2𝛼 = 𝑠𝑖𝑛−1 (2𝑥 √1 − 𝑥 2 )
⟹ 2𝑠𝑖𝑛−1 𝛼 = 𝑠𝑖𝑛−1 (2𝑥√1 − 𝑥 2 )
Similarly we can prove (i) 2𝑐𝑜𝑠 −1 𝑥 = 𝑠𝑖𝑛−1 (2𝑥√1 − 𝑥 2 ), (ii) 2𝑠𝑖𝑛−1 𝑥 = 𝑐𝑜𝑠 −1 (1 − 2𝑥 2 ),
(iii) 2𝑐𝑜𝑠 −1 𝑥 = 𝑐𝑜𝑠 −1 (2𝑥 2 − 1) (iv) 3𝑐𝑜𝑠 −1 𝑥 = 𝑐𝑜𝑠 −1 (4𝑥 3 − 3𝑥),
2𝑥
2𝑥
(v) 3𝑠𝑖𝑛−1 𝑥 = 𝑠𝑖𝑛−1 (3𝑥 − 4𝑥 3 ),(vi) 2𝑡𝑎𝑛−1 𝑥 = 𝑠𝑖𝑛−1 (1+𝑥2 ), (vii) 2𝑡𝑎𝑛−1 𝑥 = 𝑡𝑎𝑛−1 (1−𝑥 2 )
1−𝑥 2
),(ix)
1+𝑥 2
(viii) 2𝑡𝑎𝑛−1 𝑥 = 𝑐𝑜𝑠 −1 (
3𝑥−𝑥 3
),
1+3𝑥 2
3𝑡𝑎𝑛−1 𝑥 = 𝑡𝑎𝑛−1 (
𝑥 2 −1
).
2𝑥
(x) 2𝑐𝑜𝑡 −1 𝑥 = 𝑐𝑜𝑡 −1 (
6. Important substitutions required to simplify the inverse trigonometric functions.
Function
substitution
Function
√1 − 𝑥 2
𝑥 = 𝑠𝑖𝑛𝜃
√𝑥 2 − 1
𝑥 = 𝑠𝑒𝑐𝜃
√1 + 𝑥 2
𝑥 = 𝑡𝑎𝑛𝜃
√𝑥 2 + 𝑎 2
𝑥 = 𝑎𝑡𝑎𝑛𝜃
1+𝑥
1 − 𝑥2
𝑎+𝑥
1 − 𝑎𝑥
𝑥 = 𝑡𝑎𝑛𝜃,
𝑎 = 𝑡𝑎𝑛𝛼
1 − 2𝑥
𝑥 = 𝑡𝑎𝑛𝜃,
𝜋
𝑡𝑎𝑛 = 1
4
𝑥 = 𝑠𝑖𝑛𝜃
𝑥 = 𝑡𝑎𝑛𝜃
3𝑥 − 4𝑥 3
𝑥 = 𝑠𝑖𝑛𝜃
2𝑥
1 + 𝑥2
4𝑥 3 − 3𝑥
substitution
Function
substitution
Function
substitution
𝑥 = 𝑎𝑠𝑖𝑛𝜃
𝑥 = 𝑎𝑠𝑒𝑐𝜃
𝑥 = 𝑐𝑜𝑠𝜃
√𝑥 2 − 𝑎 2
𝑎−𝑥
√
𝑎+𝑥
𝑥 = 𝑎𝑐𝑜𝑠𝜃
2𝑥 √1 − 𝑥 2
𝑥 = 𝑠𝑖𝑛𝜃
2𝑥 2 − 1
𝑥 = 𝑐𝑜𝑠𝜃
2𝑥
1 − 𝑥2
3𝑥 − 𝑥 3
1 − 3𝑥 2
𝑥 = 𝑡𝑎𝑛𝜃
1 − 𝑥2
1 + 𝑥2
𝑥 = 𝑡𝑎𝑛𝜃
√𝑎 2 − 𝑥 2
√
𝑥 = 𝑐𝑜𝑠𝜃
1−𝑥
1+𝑥
𝑥 = 𝑡𝑎𝑛𝜃
7. All the inverse trigonometric functions given in each row are equivalent.
(Students are advised to prove these relations instead of by heart)
𝑥
1
𝑆𝑖𝑛−1 𝑥
√1 − 𝑥 2 𝑆𝑒𝑐 −1
𝐶𝑜𝑠 −1 √1 − 𝑥 2 𝑇𝑎𝑛−1
−1
𝐶𝑜𝑡
2
√1 − 𝑥
√1 − 𝑥 2
𝑥
−1
𝑥
1
2
𝐶𝑜𝑠 𝑥
√1 − 𝑥
𝑆𝑖𝑛−1 √1 − 𝑥 2
𝐶𝑜𝑡 −1
𝑆𝑒𝑐 −1
𝑇𝑎𝑛−1
2
√1 − 𝑥
𝑥
𝑥
𝑥
−1
1
1
𝑇𝑎𝑛 𝑥
𝑆𝑒𝑐 −1 √1 + 𝑥 2
𝑆𝑖𝑛−1
𝐶𝑜𝑡 −1
𝐶𝑜𝑠 −1
2
√1 + 𝑥 2
𝑥
√1 + 𝑥
𝑥
1
1
𝐶𝑜𝑡 −1 𝑥
−1
√1 + 𝑥 2
−1
𝐶𝑜𝑠
−1
𝑇𝑎𝑛
𝑆𝑖𝑛−1
𝑆𝑒𝑐
2
2
√1 + 𝑥
𝑥
√1 + 𝑥
𝑥
−1
1
2
1
−1
2
𝑆𝑒𝑐
𝑥
√𝑥 − 1
𝑇𝑎𝑛 √𝑥 − 1 𝐶𝑜𝑡 −1
𝐶𝑜𝑠 −1
𝑆𝑖𝑛−1
𝑥
√𝑥 2 − 1
𝑥
𝑥
𝑥
1
1
−1
𝐶𝑜𝑡 −1 √𝑥 2 − 1 𝑆𝑒𝑐 −1
𝐶𝑜𝑠 −1
𝑆𝑖𝑛−1
𝑇𝑎𝑛
√𝑥 2 − 1
√𝑥 2 − 1
𝑥
√𝑥 2 − 1
Prepared by R. VamanRao, PGT Mathematics, Kendriya Vidyalaya WARANGAL
𝐶𝑜𝑠𝑒𝑐 −1
𝐶𝑜𝑠𝑒𝑐 −1
1
𝑥
1
√1 − 𝑥 2
√1 + 𝑥 2
𝐶𝑜𝑠𝑒𝑐 −1
𝑥
−1 √
𝐶𝑜𝑠𝑒𝑐
1 + 𝑥2
𝐶𝑜𝑠𝑒𝑐 −1
𝑥
√𝑥 2 − 1
𝐶𝑜𝑠𝑒𝑐 −1 𝑥
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3. Difficult areas:
1. Students face difficulty in writing one inverse trigonometric function in terms of other trigonometric
function. The table given above will help in this regard.
2. Students face difficulty in finding right substitution for simplifying the given problem. Form of the
function and proper substitutions are given in table.
3. Students face difficulty in this chapter if they do not have the knowledge of trigonometric functions
covered in class XI. So they are advised to do all the problems of class XI from trigonometry chapter
once again.
4. Important areas of the chapter
1. A one mark question always comes from this chapter to test the knowledge of domain & range of
5𝜋
Inverse trigonometric functions. For example the value of 𝑆𝑖𝑛−1 (𝑠𝑖𝑛 ( 3 ))cannot be
5𝜋
because it
3
is
not in the range of𝑆𝑖𝑛−1 ( ). A good practice of such type of problems may be given.
1
2. One four marks question always comes from this chapter of the type (i) Prove 2𝑡𝑎𝑛−1 2 =
4
3
𝑡𝑎𝑛−1 or (ii) simplify 𝑠𝑖𝑛−1 (
2𝑥
√1−𝑥 2
).
5. Practice problems:
−1
√2
1. write the value of 𝑐𝑜𝑠 −1 ( ).
3
5
63
2. Show that 𝑠𝑖𝑛−1 (5) + 𝑐𝑜𝑠 −1 (13) = 𝑠𝑖𝑛−1 (65)
3𝑥−𝑥 3
3. Simplify 𝑡𝑎𝑛−1 (1−3𝑥2 )
3
4
3
5
8
19
4. Prove that 𝑡𝑎𝑛−1 ( ) + 𝑡𝑎𝑛−1 ( ) − 𝑡𝑎𝑛−1 ( ) =
5. Find 𝑡𝑎𝑛−1 (tan
𝜋
4
7𝜋
)
6
√1+𝑠𝑖𝑛𝑥+√1−𝑠𝑖𝑛𝑥
).
√1+𝑠𝑖𝑛𝑥−√1−𝑠𝑖𝑛𝑥
6. Simplify 𝑐𝑜𝑡 −1 (
7. Prove that 𝑐𝑜𝑡 −1 (13) + 𝑐𝑜𝑡 −1 (21) + 𝑐𝑜𝑡 −1 (−8) = 𝜋
8. Find 𝑆𝑖𝑛 𝑥 𝑖𝑓 𝑐𝑜𝑡 −1 (√𝑐𝑜𝑠 ∝) − 𝑡𝑎𝑛−1 (√𝑐𝑜𝑠 ∝) = 𝑥, 𝑥 ≥ 0
1
√5
9. Evaluate 𝑡𝑎𝑛 [2 𝑐𝑜𝑠 −1 ( 3 )]
𝜋
10. If 𝑡𝑎𝑛−1 (𝑥) + 𝑡𝑎𝑛−1 (𝑥) = 4 , find the value of x + y + xy.
11. Evaluate 𝑐𝑜𝑠 [ 𝑐𝑜𝑠 −1 (−
𝜋
√3
)+ ]
2
6
12. Solve for ‘x’ : tan−1(√𝑥(𝑥 + 1)) + Sin−1(√𝑥 2 + 𝑥 + 1) =
𝑎𝑏+1
𝑏𝑐+1
𝜋
2
𝑐𝑎+1
13. Prove that : cot −1 ( 𝑎−𝑏 ) + cot −1 ( 𝑏−𝑐 ) + cot −1 ( 𝑐−𝑎 ) = 0
1+𝑥 2
14. Prove that : cos(tan−1 (sin(cot −1 𝑥))) = √2+𝑥2
Prepared by R. VamanRao, PGT Mathematics, Kendriya Vidyalaya WARANGAL
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CHAPTER 1
RELATIONS AND FUNCTIONS
1 . Let f: R →R be defined by f(x) = x |𝑥|, State whether the function f(x) is
onto.
2. Letbe the binary operations on Zgiven by a * b = a + b + 1 ∀a, b∈.Find
the identify element for * on Z, if any.
3. State with reason whether the function f: X→Y have inverse,
f(x)=
1
𝑋
where
.and X=Q-o, Y=Q .
4. Let Y = {n2: n∈N} be a subset of N and let “f” be a function f : N →Y
defined as f(x) = x2.
Show that “f” is invertible and find inverse of “f”.
5. Show that the function f : N →N given by f(x) = x – (–1)x is bijective.
6. If f be the greatest integer function and g be the absolute value function;
find the value of (fog)(-3/2) + (gof)(4/3).
7.Consider the mapping f :[0,2] → [0,2] defined by f(x)= √4 − 𝑥 2 Show that f is
invertible and hence find f-1.
8. Give examples of two functions f : N →N and g :Z →Z such that gof is
injective but g is not injective
9..Give examples of two functions f:N →N and g: N →N such that gof is onto but f
is not onto.
10..Let f: R- -3/5 →R be a function defined as f(x) =
2𝑥
, find the inverse
5𝑥+3
of f.
11. Show that the relation R defined by (a, b) R (c, d)⇒a + d = b + c on the
set NXN is an equivalence relation.
12.Let Q+ be the set of all positive rational numbers.
Show that the operation * on Q+ defined by a*b = ½ (a+b) is a binary
operation.
Show that is commutative.
Show that is not associative.
13. Let A= N X N. Let * be a binary operation on A defined by (a,b)*(c,d) = (ad
+bc, bd)∀a,b,c,d ∈N. Show that (i) is commutative (ii) is associative (iii)
identity element w.r.t. does not exist.
14. Draw the graph of the function f(x) = x2 on R and show that it is not
invertible. Restrict its domain suitably so that f-1 may exist, find f-1 and
draw its graph.
Prepared by R. VamanRao, PGT Mathematics, Kendriya Vidyalaya WARANGAL
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