Chapter 2 - Inverse Trigonometric Functions
NCERT Solutions for Class 12 Science Math
Exercise No. 2.2
Question 1:
Prove ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_m5f173286.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_m5f173286.gif)
ANSWER:
To prove: ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_m5f173286.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_m5f173286.gif)
Let x = sinθ. Then, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_43d4773b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_43d4773b.gif)
We have,
R.H.S. =![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_med71555.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_med71555.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_m12ab92b3.gif)
= 3θ
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_m4406dcf8.gif)
= L.H.S.
Page No 47:
Question 2:
Prove ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8237/Chapter%202_html_m1e80cc21.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8237/Chapter%202_html_m1e80cc21.gif)
ANSWER:
To prove:![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8237/Chapter%202_html_m1e80cc21.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8237/Chapter%202_html_m1e80cc21.gif)
Let x = cosθ. Then, cos−1 x =θ.
We have,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8237/Chapter%202_html_6f7056ef.gif)
Page No 47:
Question 3:
Prove ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8238/Chapter%202_html_m7961d769.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8238/Chapter%202_html_m7961d769.gif)
ANSWER:
To prove:![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8238/Chapter%202_html_m7961d769.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8238/Chapter%202_html_m7961d769.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8238/Chapter%202_html_m2b8918f0.gif)
Page No 47:
Question 4:
Prove ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8239/Chapter%202_html_3c045b5e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8239/Chapter%202_html_3c045b5e.gif)
ANSWER:
To prove: ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8239/Chapter%202_html_3c045b5e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8239/Chapter%202_html_3c045b5e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8239/Chapter%202_html_261ec9f5.gif)
Page No 47:
Question 5:
Write the function in the simplest form:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8240/Chapter%202_html_13d7e339.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8240/Chapter%202_html_7c792e6d.gif)
Page No 47:
Question 6:
Write the function in the simplest form:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8241/Chapter%202_html_44000ff6.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8241/Chapter%202_html_44000ff6.gif)
Put x = cosec θ ⇒ θ = cosec−1 x
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8241/Chapter%202_html_76ec630a.gif)
Page No 47:
Question 7:
Write the function in the simplest form:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8242/Chapter%202_html_44231546.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8242/Chapter%202_html_4c718ec7.gif)
Question 8:
Write the function in the simplest form:
Write the function in the simplest form:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8243/Chapter%202_html_m4c6bdcfa.gif)
ANSWER:
Page No 48:
Question 9:
Write the function in the simplest form:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8244/Chapter%202_html_m87c9839.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8244/Chapter%202_html_m6e536503.gif)
Page No 48:
Question 10:
Write the function in the simplest form:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8245/Chapter%202_html_494cb76b.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8245/Chapter%202_html_7e7fb12f.gif)
Page No 48:
Question 11:
Find the value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8246/Chapter%202_html_64df0e7a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8246/Chapter%202_html_64df0e7a.gif)
ANSWER:
Let
. Then,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8246/Chapter%202_html_mefaf402.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8246/Chapter%202_html_m26c9b49.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8246/Chapter%202_html_mefaf402.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8246/Chapter%202_html_10923d49.gif)
Page No 48:
Question 12:
Find the value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8247/Chapter%202_html_m1a481954.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8247/Chapter%202_html_m1a481954.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8247/Chapter%202_html_52443149.gif)
Page No 48:
Question 13:
Find the value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8248/Chapter%202_html_m20f9e382.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8248/Chapter%202_html_m20f9e382.gif)
ANSWER:
Let x = tan θ. Then, θ = tan−1 x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8248/Chapter%202_html_c7f5408.gif)
Let y = tan Φ. Then, Φ = tan−1 y.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8248/Chapter%202_html_245cd23b.gif)
Page No 48:
Question 14:
If
, then find the value of x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8249/Chapter%202_html_4a5af6ac.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8249/Chapter%202_html_m2360a54e.gif)
On squaring both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8249/Chapter%202_html_mdbee6ce.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8249/Chapter%202_html_6fbad75c.gif)
Hence, the value of x is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8249/Chapter%202_html_m1f05a0cd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8249/Chapter%202_html_m1f05a0cd.gif)
Page No 48:
Question 15:
If
, then find the value of x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8250/Chapter%202_html_m1a20e5ca.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8250/Chapter%202_html_m36820ccf.gif)
Hence, the value of x is ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8250/Chapter%202_html_m76605374.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8250/Chapter%202_html_m76605374.gif)
Page No 48:
Question 16:
Find the values of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m1df80c55.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m1df80c55.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m1df80c55.gif)
We know that sin−1 (sin x) = x if
, which is the principal value branch of sin−1x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_426a3638.gif)
Here,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m1ed0bee4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m1ed0bee4.gif)
Now,
can be written as:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m1df80c55.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m56e77f67.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m459728e3.gif)
Page No 48:
Question 17:
Find the values of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_m496170f3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_m496170f3.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_m496170f3.gif)
We know that tan−1 (tan x) = x if
, which is the principal value branch of tan−1x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_m448424b2.gif)
Here,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_m75a4fa3b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_m75a4fa3b.gif)
Now,
can be written as:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_m496170f3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_75479362.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_155e1c32.gif)
Page No 48:
Question 18:
Find the values of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8253/Chapter%202_html_626d4d8e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8253/Chapter%202_html_626d4d8e.gif)
ANSWER:
Let
. Then,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8253/Chapter%202_html_m6b228fa3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8253/Chapter%202_html_7835c198.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8253/Chapter%202_html_m6b228fa3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8253/Chapter%202_html_2b0e4f31.gif)
Page No 48:
Page No 48:
Question 20:
Find the values of
is equal to
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_607ccadd.gif)
(A)
(B)
(C)
(D)1
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_eeecab0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_33f00ded.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_682bd651.gif)
ANSWER:
Let
. Then, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_m7ab2232f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_m2bc5b727.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_m7ab2232f.gif)
We know that the range of the principal value branch of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_2d4d89cf.gif)
∴![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_m25255b6c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_m25255b6c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_68b41bac.gif)
The correct answer is D.
Page No 48:
Question 21:
Find the values of
is equal to
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_m791a90d7.gif)
(A)π (B)
(C) 0 (D)![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_m465f17c8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_m4d28d5df.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_m465f17c8.gif)
ANSWER:
Let
. Then,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_27471169.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_7f97e1a5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_27471169.gif)
We know that the range of the principal value branch of![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_1ad054a8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_1ad054a8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_622d697f.gif)
Let
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_m631d3ca8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_m514fe183.gif)
The range of the principal value branch of![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_53b51094.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_53b51094.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_754fcfa8.gif)
The correct answer is B.