Chapter 2 - Inverse Trigonometric FunctionsNCERT Solutions for Class 12 Science Math
Exercise No. 2.1
Question 1:
Find the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6152/Chapter%202_html_m34ac5c0c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6152/Chapter%202_html_m34ac5c0c.gif)
ANSWER:
Let sin-1
Then sin y = ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6152/Chapter%202_html_7f23d2b8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6152/Chapter%202_html_m76493cdd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6152/Chapter%202_html_7f23d2b8.gif)
We know that the range of the principal value branch of sin−1 is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6152/Chapter%202_html_630b559d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6152/Chapter%202_html_m52a394b8.gif)
Therefore, the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6152/Chapter%202_html_m10af7c0f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6152/Chapter%202_html_m10af7c0f.gif)
Page No 41:
Question 2:
Find the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6155/Chapter%202_html_m2199c57.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6155/Chapter%202_html_m2199c57.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6155/Chapter%202_html_21c6dd3e.gif)
We know that the range of the principal value branch of cos−1 is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6155/Chapter%202_html_11c3782d.gif)
Therefore, the principal value of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6155/Chapter%202_html_44d8e3cb.gif)
Page No 41:
Question 3:
Find the principal value of cosec−1 (2)
ANSWER:
Let cosec−1 (2) = y. Then, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6157/Chapter%202_html_44bfd2fc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6157/Chapter%202_html_44bfd2fc.gif)
We know that the range of the principal value branch of cosec−1 is ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6157/Chapter%202_html_48c56e9c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6157/Chapter%202_html_48c56e9c.gif)
Therefore, the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6157/Chapter%202_html_m6570ab7f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6157/Chapter%202_html_m6570ab7f.gif)
Page No 41:
Question 4:
Find the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6159/Chapter%202_html_m43312614.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6159/Chapter%202_html_m43312614.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6159/Chapter%202_html_30b6049c.gif)
We know that the range of the principal value branch of tan−1 is ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6159/Chapter%202_html_m5dee6bcf.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6159/Chapter%202_html_m5dee6bcf.gif)
Therefore, the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6159/Chapter%202_html_55e94359.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6159/Chapter%202_html_55e94359.gif)
Page No 41:
Question 5:
Find the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6161/Chapter%202_html_f32d608.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6161/Chapter%202_html_f32d608.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6161/Chapter%202_html_49ef2e20.gif)
We know that the range of the principal value branch of cos−1 is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6161/Chapter%202_html_3c9c7a14.gif)
Therefore, the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6161/Chapter%202_html_2729478b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6161/Chapter%202_html_2729478b.gif)
Page No 41:
Question 6:
Find the principal value of tan−1 (−1)
ANSWER:
Let tan−1 (−1) = y. Then, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6164/Chapter%202_html_m587828a4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6164/Chapter%202_html_m587828a4.gif)
We know that the range of the principal value branch of tan−1 is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6164/Chapter%202_html_m6f71dbaf.gif)
Therefore, the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6164/Chapter%202_html_m7edf9658.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6164/Chapter%202_html_m7edf9658.gif)
Page No 42:
Question 7:
Find the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6169/Chapter%202_html_m36a0c374.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6169/Chapter%202_html_m36a0c374.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6169/Chapter%202_html_m701b1a3c.gif)
We know that the range of the principal value branch of sec−1 is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6169/Chapter%202_html_m63fd3adb.gif)
Therefore, the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6169/Chapter%202_html_490890b8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6169/Chapter%202_html_490890b8.gif)
Page No 42:
Question 8:
Find the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6173/Chapter%202_html_1e6e5b08.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6173/Chapter%202_html_1e6e5b08.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6173/Chapter%202_html_m61fa0f47.gif)
We know that the range of the principal value branch of cot−1 is (0,π) and
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6173/Chapter%202_html_4483fc22.gif)
Therefore, the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6173/Chapter%202_html_1fd9c850.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6173/Chapter%202_html_1fd9c850.gif)
Page No 42:
Question 9:
Find the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6177/Chapter%202_html_3f88ddef.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6177/Chapter%202_html_3f88ddef.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6177/Chapter%202_html_59f0c28b.gif)
We know that the range of the principal value branch of cos−1 is [0,π] and
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6177/Chapter%202_html_1f6e0450.gif)
Therefore, the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6177/Chapter%202_html_5591b560.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6177/Chapter%202_html_5591b560.gif)
Page No 42:
Question 10:
Find the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6189/Chapter%202_html_m1ceae9b6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6189/Chapter%202_html_m1ceae9b6.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6189/Chapter%202_html_41667908.gif)
We know that the range of the principal value branch of cosec−1 is ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6189/Chapter%202_html_384482b9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6189/Chapter%202_html_384482b9.gif)
Therefore, the principal value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6189/Chapter%202_html_m4d1d88dc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6189/Chapter%202_html_m4d1d88dc.gif)
Page No 42:
Question 11:
Find the value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6194/Item%2011_html_m5f2a02e7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6194/Item%2011_html_m5f2a02e7.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6194/Item%2011_html_7da7da07.gif)
Page No 42:
Question 12:
Find the value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6198/Chapter%202_html_6b7da109.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6198/Chapter%202_html_6b7da109.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/6198/Chapter%202_html_m773dc34b.gif)
Page No 42:
Question 13:
Find the value of if sin−1 x = y, then
(A)
(B)![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8234/Chapter%202_html_3410905f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8234/Chapter%202_html_m73e5ee59.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8234/Chapter%202_html_3410905f.gif)
(C)
(D) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8234/Chapter%202_html_m71e7aca8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8234/Chapter%202_html_m39d75a48.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8234/Chapter%202_html_m71e7aca8.gif)
ANSWER:
It is given that sin−1 x = y.
We know that the range of the principal value branch of sin−1 is ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8234/Chapter%202_html_1db77eaa.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8234/Chapter%202_html_1db77eaa.gif)
Therefore,
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8234/Chapter%202_html_3410905f.gif)
Page No 42:
Question 14:
Find the value of
is equal to
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8235/Chapter%202_html_1a0f6b43.gif)
(A)π(B)
(C)
(D) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8235/Chapter%202_html_552167fc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8235/Chapter%202_html_m912018c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8235/Chapter%202_html_m4e8d241e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8235/Chapter%202_html_552167fc.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8235/Chapter%202_html_612833b1.gif)