Post on 12-Jan-2022
Exercise 2.1 Page No: 41
Find the principal values of the following:
1. ๐ฌ๐ข๐งโ๐ (โ๐
๐)
2.
3. Cosec -1 (2)
4.
5.
6. tan-1(-1)
7.
8.
9.
10.
Solution 1: Consider y = sinโ1 (โ1
2)
Solve the above equation, we have sin y = -1/2 We know that sin ฯ/6 = ยฝ So, sin y = - sin ฯ/6
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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Since range of principle value of sin-1 is
Principle value of sinโ1 (โ1
2) is - ฯ/6.
Solution 2:
Let y =
Cos y = cos ฯ/6 (as cos ฯ/6 = โ3 / 2 )
y = ฯ/6
Since range of principle value of cos-1 is [0, ฯ]
Therefore, Principle value of is ฯ/6
Solution 3: Cosec -1 (2)
Let y = Cosec -1 (2)
Cosec y = 2
We know that, cosec ฯ /6 = 2
So Cosec y = cosec ฯ /6
Since range of principle value of cosec-1 is
Therefore, Principle value of Cosec -1 (2) is ฮ /6.
Solution 4:
Let y =
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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tan y = - tan ฯ/3
or tan y = tan (-ฯ/3)
Since range of principle value of tan-1 is
Therefore, Principle value of is โ ฯ/3.
Solution 5:
y =
cos y = -1/2
cos y = cos(ฯ โ ฯ/3) = cos (2ฯ/3)
Since principle value of cos-1 is [0, ฯ]
Therefore, Principle value of is 2ฯ/3.
Solution 6: tan-1(-1)
Let y = tan-1(-1)
tan (y) = -1
tan y = -tan ฯ/4
Since principle value of tan-1 is
Therefore, Principle value of tan-1(-1) is - ฯ/4.
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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Solution 7:
y =
sec y = 2/โ3
Since principle value of sec-1 is [0, ฯ]
Therefore, Principle value of is ฯ/6
Solution 8:
y =
cot y = โ3
cot y = ฯ/6
Since principle value of cot-1 is [0, ฯ]
Therefore, Principle value of is ฯ/6.
Solution 9:
Let y =
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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Since principle value of cos-1 is [0, ฯ]
Therefore, Principle value of is 3 ฯ/ 4.
Solution 10.
Ley y =
Since principle value of cosec-1 is
Therefore, Principle value of is โ ฯ/4
Find the values of the following:
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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13. If sinโ1 x = y, then
(A) 0 โค y โค ฯ
(B) โ๐
๐โค ๐ฒ โค
๐
๐
(C) 0 < y < ฯ
(D) โ๐
๐< ๐ฒ <
๐
๐
14. tanโ1 ( โ๐) - sec -1 (-2) is equal to
(A) ฯ
(B) - ฯ/3
(C) ฯ/3
(D) 2 ฯ/3
Solution 11.
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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Solution 12:
Solution 13: Option (B) is correct.
Given sinโ1 x = y,
The range of the principle value of sin-1 is
Therefore, โฯ
2โค y โค
ฯ
2
Solution 14:
Option (B) is correct.
tanโ1 ( โ3) - sec -1 (-2) = tan-1 (tan ฯ/3) โ sec-1 (-sec ฯ/3)
= ฯ/3 - sec-1 (sec (ฯ - ฯ/3))
= ฯ/3 - 2ฯ/3 = - ฯ/3
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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Exercise 2.2 Page No: 47
Prove the following
1.
Solution:
(Use identity: )
Let then
Now, RHS
= sinโ1(3๐ฅ โ 4๐ฅ3)
= sinโ1(3 sin ฮธ โ 4 sin3 ฮธ)
= sinโ1(sin 3 ฮธ)
= 3 ฮธ
= 3 sin-1 x
= LHS
Hence Proved
2.
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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Solution:
Using identity:
Put x = cos ฮธ
ฮธ = cos-1 (x)
Therefore, cos 3 ฮธ = 4x3 โ 3x
RHS:
= cos-1 (cos 3 ฮธ)
= 3 ฮธ
= 3 cos-1 (x)
= LHS
Hence Proved.
3.
Solution:
Using identity:
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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LHS =
= tan -1 (125/250)
= tan -1 (1/2)
= RHS
Hence Proved
4.
Solution:
Use identity:
LHS
=
=
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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= tan-1(4/3) + tan-1(1/7)
Again using identity:
We have,
= ๐ก๐๐โ1(28+3
21โ4 )
= tan-1 (31/17)
RHS
Write the following functions in the simplest form:
5.
Solution:
Letโs say x = tan ฮธ then ฮธ = tan -1 x
We get,
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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This is simplest form of the function.
6.
Solution:
Let us consider, x = sec ฮธ, then ฮธ = sec-1 x
= tan-1(cot ฮธ)
= tan-1 tan(ฯ/2 - ฮธ)
= (ฯ/2 - ฮธ)
= ฯ/2 โ sec-1 x
This is simplest form of the given function.
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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7.
Solution:
8.
Solution:
Divide numerator and denominator by cos x, we have
๐๐๐โ๐(
๐๐๐(๐)
๐๐๐(๐)โ
๐๐๐(๐)
๐๐๐(๐)
๐๐๐(๐)
๐๐๐(๐)+
๐๐๐(๐)
๐๐๐(๐)
)
= ๐๐๐โ๐(๐ โ
๐๐๐(๐)
๐๐๐(๐)
๐ +๐๐๐(๐)
๐๐๐(๐)
)
=
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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= tan -1 tan(ฯ/4 โ x)
= ฯ/4 โ x
9.
Solution:
Put x = a sin ฮธ, which implies sin ฮธ = x/a and ฮธ = sin-1(x/a)
Substitute the values into given function, we get
= tan-1(tan ฮธ)
= ฮธ
= sin-1(x/a)
10.
Solution: After dividing numerator and denominator by a^3 we have
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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Put x/a = tan ฮธ and ฮธ = tan-1(x/a)
=
= tan-1 (tan 3 ฮธ)
= 3 ฮธ
= 3 tan-1(x/a)
Find the values of each of the following:
11.
Solution:
= tan-1 (2 cos ฯ/3)
= tan-1(2 x ยฝ)
= tan-1 (1)
= tan-1 (tan (ฯ/4))
= ฯ/4
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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12. cot (tanโ1a + cotโ1a)
Solution:
cot (tanโ1a + cotโ1a) = cot ฯ/2 = 0
Using identity: tanโ1a + cotโ1a = ฯ/2
13.
Solution:
Put x = tan ฮธ and y = tan ะค, we have
= tan1/2[sin-1 sin 2 ฮธ + cos-1 cos 2 ะค]
= tan (1/2) [2 ฮธ + 2 ะค]
= tan (ฮธ + ะค)
=
= (x+y) / (1-xy)
14. If , then find the value of x.
Solution: We know that, sin 90 degrees = sin ฯ/2 = 1
So, given equation turned as,
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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Using identity: = ฯ/2
We have,
Which implies, the value of x is 1/5.
15. If , then find the value of x.
Solution:
We have reduced the given equation using below identity:
or
or
๐๐ 2๐ฅ2 โ 4
๐ฅ2 โ 4 โ ๐ฅ2 + 1= tan (
๐
4)
or (2x^2 - 4)/-3 = 1
or 2x^2 = 1
or x = ยฑ 1
โ2
The value of x is either 1
โ2๐๐ โ
1
โ2
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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Find the values of each of the expressions in Exercises 16 to 18.
16. ๐ฌ๐ข๐งโ๐(๐ฌ๐ข๐ง (๐๐
๐))
Solution:
Given expression is sinโ1(sin (2๐
3))
First split 2๐
3๐๐
(3๐โ๐)
3 ๐๐ ๐ โ
๐
3
After substituting in given we get,
sinโ1(sin (2๐
3)) = sinโ1(sin (๐ โ
๐
3)) =
๐
3
Therefore, the value of sinโ1 (sin (2๐
3)) is
๐
3
17. ๐๐๐โ๐(๐ญ๐๐ง (๐๐
๐))
Solution:
Given expression is ๐ก๐๐โ1(tan (3๐
4))
First split 3๐
4๐๐
(4๐โ๐)
4 ๐๐ ๐ โ
๐
4
After substituting in given we get,
๐ก๐๐โ1(tan (3๐
4)) = tanโ1(tan (๐ โ
๐
4 )) = โ
๐4
The value of ๐ก๐๐โ1(tan (3๐
4)) is
โ๐
4.
18. ๐ญ๐๐ง (๐๐๐โ๐ (๐
๐) + ๐๐จ๐ญโ๐ ๐
๐)
Solution:
Given expression is tan (๐ ๐๐โ1 (3
5) + cotโ1 3
2)
Putting, ๐ ๐๐โ1 (3
5) = ๐ฅ ๐๐๐ cotโ1(
3
2) = ๐ฆ
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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Or sin(x) = 3/5 and cot y = 3/2
Now, sin(x) = 3/5 => cos x = โ1 โ sin2 ๐ฅ = 4/5 and sec x = 5/4
(using identities: cos x = โ1 โ sin2 ๐ฅ and sec x = 1/cos x)
Again, tan x = โsec2 ๐ฅ โ 1 = โ25
16โ 1 = ยพ and tan y = 1/cot(y) = 2/3
Now, we can write given expression as,
tan (๐ ๐๐โ1 (3
5) + cotโ1 3
2) = tan(x + y)
=
= 17/6
19. ๐๐จ๐ฌโ๐(๐๐จ๐ฌ๐๐
๐) is equal to
(A) 7ฯ/6 (B) 5 ฯ/6 (C) ฯ/3 (D) ฯ/6
Solution:
Option (B) is correct.
Explanation:
cosโ1(cos7ฯ
6) = cosโ1(cos (2ฯ โ
7ฯ
6)
(As cos (2 ฯ โ A) = cos A)
Now 2ฯ โ 7ฯ
6=
12ฯโ7ฯ
6=
5ฯ
6
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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20. is equal to
(A) ยฝ (B) 1/3 (C) ยผ (D) 1
Solution:
Option (D) is correct
Explanation:
First solve for: sinโ1 (โ1
2)
= - ฯ/6
Again,
= sin(ฯ/2)
= 1
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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21. tan-1 โ๐ โ cot-1 ( -โ๐) is equal to
(A) ฯ (B) -ฯ/2 (C) 0 (D) 2โ๐
Solution:
Option (B) is correct.
Explanation:
tan-1 โ3 โ cot-1 ( -โ3) can be written as
=๐
3โ (๐ โ
๐
6)
= ๐
3โ
5๐
6
=โ3๐
6
= - ๐/2
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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Miscellaneous Exercise Page No: 51
Find the value of the following:
1. ๐๐จ๐ฌโ๐(๐๐จ๐ฌ๐๐๐
๐)
Solution:
First solve for, cos13๐
6= cos(2 ๐ +
๐
6) = cos
๐
6
Now: cosโ1(cos13๐
6) = cosโ1(cos
๐
6) =
๐
6 โ [0, ฯ]
[As cos-1 cos(x) = x if x โ [0, ฯ] ]
So the value of cosโ1(cos13๐
6) is
๐
6.
2. ๐๐๐โ๐(๐ญ๐๐ง๐๐
๐)
Solution:
First solve for, tan7๐
6= tan(๐ +
๐
6) = tan
๐
6
Now: ๐ก๐๐โ1(tan7๐
6) = tanโ1(tan
๐
6) =
๐
6 โ (-ฯ/2, ฯ/2)
[As tan-1 tan(x) = x if x โ (-ฯ/2, ฯ/2) ]
So the value of ๐ก๐๐โ1(tan7๐
6) is
๐
6.
3. Prove that ๐ ๐ฌ๐ข๐งโ๐ ๐
๐= ๐ญ๐๐งโ๐ ๐๐
๐
Solution: Step 1: Find the value of cos x and tan x
Let us considersinโ1 3
5= ๐ฅ, then sin x = 3/5
So, cos x = โ1 โ sin2 ๐ฅ = โ1 โ (3
5)
2
= 4/5
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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tan x = sin x/ cos x = ยพ
Therefore, x = tan-1 (3/4), substitute the value of x,
sinโ1 3
5= tanโ1 (
3
4) โฆ..(1)
Step 2: Solve LHS
2 sinโ13
5= 2 tanโ1
3
4
Using identity: 2tan-1 x = tanโ1 = tanโ1(2๐ฅ
1โ๐ฅ2), we get
= tanโ1(2(
3
4)
1โ(3
4)
2)
= tan-1(24/7)
= RHS
Hence Proved.
4. Prove that ๐ฌ๐ข๐งโ๐ ๐
๐๐+ ๐ฌ๐ข๐งโ๐ ๐
๐= ๐ญ๐๐งโ๐ ๐๐
๐๐
Solution:
Let sinโ1 (8
17) = ๐ฅ then sin x = 8/17
Again, cos x = โ1 โ sin2 ๐ฅ = โ1 โ64
289 = 15/17
And tan x = sin x / cos x = 8/ 15
Again,
Let sinโ1 (3
5) = ๐ฆ then sin y = 3/5
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
NCERT Solutions
Again, cos y = โ1 โ sin2 ๐ฆ = โ1 โ9
25 = 4/5
And tan y = sin y / cos y = ยพ
Solve for tan(x + y), using below identity,
= 32+45
60โ24
= 77/36
This implies x + y = tan-1(77/36)
Substituting the values back, we have
sinโ1 8
17+ sinโ1 3
5= tanโ1 77
36 (Proved)
5. Prove that ๐๐จ๐ฌโ๐ (๐
๐) + ๐๐จ๐ฌโ๐ (
๐๐
๐๐) = ๐๐จ๐ฌโ๐ (
๐๐
๐๐)
Solution:
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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Solve the expression, Using identity: cos (ฮธ + ฯ) = cosฮธ cos ฯ - sinฮธ sin ฯ
= 4/5 x 12/13 โ 3/5 x 5/13
= (48-15)/65
= 33/65
This implies cos (ฮธ + ฯ) = 33/65
or ฮธ + ฯ = cos-1 (33/65)
Putting back the value of ฮธ and ฯ, we get
cosโ1 (4
5) + cosโ1 (
12
13) = cosโ1 (
33
65)
Hence Proved.
6. Prove that ๐๐จ๐ฌโ๐ (๐๐
๐๐) + ๐ฌ๐ข๐งโ๐ (
๐
๐) = ๐ฌ๐ข๐งโ๐ (
๐๐
๐๐)
Solution:
Solve the expression, Using identity: sin (ฮธ + ฯ) = sin ฮธ cos ฯ + cos ฮธ sin ฯ
= 12/13 x 3/5 + 12/13 x 3/5
= (20+36)/65
= 56/65
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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or sin (ฮธ + ฯ) = 56/65
or ฮธ + ฯ) = sin -1 56/65
Putting back the value of ฮธ and ฯ, we get
cosโ1 (12
13) + sinโ1 (
3
5) = sinโ1 (
56
65)
Hence Proved.
7. Prove that ๐ญ๐๐งโ๐ (๐๐
๐๐) = ๐ฌ๐ข๐งโ๐ (
๐
๐๐) + ๐๐จ๐ฌโ๐ (
๐
๐)
Solution:
Solve the expression, Using identity:
=
5
12+
4
3
1โ5
12ร
4
3
= 63/16
(ฮธ + ฯ) = tan-1 (63/16)
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
NCERT Solutions
Putting back the value of ฮธ and ฯ, we get
tanโ1 (63
16) = sinโ1 (
5
13) + cosโ1 (
3
5)
Hence Proved.
8. Prove that ๐ญ๐๐งโ๐ (๐
๐) + ๐ญ๐๐งโ๐ (
๐
๐) + ๐ญ๐๐งโ๐ (
๐
๐) + ๐ญ๐๐งโ๐ (
๐
๐) =
๐
๐
Solution:
LHS = (tanโ1 (1
5) + tanโ1 (
1
7)) + ( tanโ1 (
1
3) + tanโ1 (
1
8) )
Solve above expressions, using below identity:
= tanโ1(1
5+
1
7
1โ1
5ร
1
7
) + tanโ1(1
3+
1
8
1โ1
3ร
1
8
)
After simplifying, we have
= tan-1 (6/17) + tan-1 (11/23)
Again, applying the formula, we get
After simplifying,
= tan-1(325/325)
= tan-1 (1)
= ฯ/4
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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9. Prove that ๐ญ๐๐งโ๐ โ๐ = ๐
๐๐๐จ๐ฌโ๐ ๐โ๐
๐+๐ , x โ (0, 1)
Solution:
Let tanโ1 โ๐ฅ = ฮธ , then โ๐ฅ = tan ฮธ
Squaring both the sides
tan2 ฮธ = x
Now, substitute the value of x in 1
2cosโ1 1โ๐ฅ
1+๐ฅ , we get
= ยฝ cos-1 (cos 2 ฮธ)
= ยฝ (2 ฮธ)
= ฮธ
= tanโ1 โ๐ฅ
10. Prove that cot-1 (โ๐+๐ฌ๐ข๐ง ๐ + โ๐โ๐ฌ๐ข๐ง ๐โ๐+๐ฌ๐ข๐ง ๐ โ โ๐โ๐ฌ๐ข๐ง ๐
) = ๐๐, x โ (0, ฯ/4)
Solution: We can write 1+ sin x as,
And
LHS:
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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= cot-1(2 cos (
๐ฅ
2)
2 sin(๐ฅ
2))
= cot-1 (cot (x/2)
= x/2
11. Prove that ๐ญ๐๐งโ๐(โ๐+๐ โโ๐โ๐
โ๐+๐ + โ๐โ๐) =
๐
๐โ
๐
๐๐๐จ๐ฌโ๐ ๐ , โ
๐
โ๐ โค ๐ โค ๐
[Hint: Put x = cos 2 ฮธ]
Solution:
Divide each term by โ2 cos ฮธ
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
NCERT Solutions
= RHS
Hence proved
12. Prove that ๐๐
๐โ
๐
๐๐ฌ๐ข๐งโ๐ ๐
๐=
๐
๐๐ฌ๐ข๐งโ๐ ๐โ๐
๐
Solution:
LHS = 9๐
8โ
9
4sinโ1 1
3
โฆโฆ.(1)
(Using identity: )
Let ฮธ = cos-1 (1/3), so cos ฮธ = 1/3
As
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
NCERT Solutions
Using equation (1), 9
4sinโ1 2โ2
3
Which is right hand side of the expression.
Solve the following equations:
13. 2tan-1 (cos x) = tan-1 (2 cosec x)
Solution:
Cot x = 1
x = ฯ/4
14. Solve
Solution:
Put x = tan ฮธ
This implies
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
NCERT Solutions
ฯ/4 โ ฮธ = ฮธ /2
or 3ฮธ /2 = ฯ/4
ฮธ = ฯ/6
Therefore, x = tan ฮธ = tan ฯ/6 = 1/โ3
15. is equal to
Solution:
Option (D) is correct.
Explanation:
Again, Letโs say
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
NCERT Solutions
This implies,
Which shows,
16. then x is equal to
(A) 0, ยฝ (B) 1, ยฝ (C) 0 (D) ยฝ
Solution:
Option (C) is correct.
Explanation:
Now,
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
NCERT Solutions
(As x = sin ฮธ)
After simplifying, we get
x(2x โ 1) = 0
x = 0 or 2x โ 1 = 0
x = 0 or x = ยฝ
Equation is not true for x = ยฝ. So the answer is x = 0.
17. is equal to
(A) ฯ/2 (B) ฯ/3 (C) ฯ/4 (D) -3 ฯ/4
Solution:
Option (C) is correct.
Explanation:
Given expression can be written as,
Class - XII _ Maths Ch. 2 - Inverse-Trigonometric-Functions
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