NCERT Solutions for Class 11 Maths Chapter 3 – Trigonometric Functions Miscellaneous Exercise
Page No 81:
Question 1:
Prove that: ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5532/Chapter%203_html_48fa2043.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5532/Chapter%203_html_48fa2043.gif)
Answer:
L.H.S.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5532/Chapter%203_html_m151c395d.gif)
= 0 = R.H.S
Question 2:
Prove that: (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0
Answer:
L.H.S.
= (sin 3x + sin x) sin x + (cos 3x – cos x) cos x
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5533/Chapter%203_html_m679e0c22.gif)
= RH.S.
Page No 82:
Question 3:
Prove that: ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5534/Chapter%203_html_m69c0eade.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5534/Chapter%203_html_m69c0eade.gif)
Answer:
L.H.S. = ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5534/Chapter%203_html_m70af887f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5534/Chapter%203_html_m70af887f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5534/Chapter%203_html_6463d4ae.gif)
Question 4:
Prove that: ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5535/Chapter%203_html_m7a7f0bd5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5535/Chapter%203_html_m7a7f0bd5.gif)
Answer:
L.H.S. = ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5535/Chapter%203_html_m5ebce28c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5535/Chapter%203_html_m5ebce28c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5535/Chapter%203_html_7b3296e6.gif)
Question 5:
Prove that: ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5536/Chapter%203_html_21ab363e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5536/Chapter%203_html_21ab363e.gif)
Answer:
It is known that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5536/Chapter%203_html_e068b49.gif)
∴L.H.S. = ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5536/Chapter%203_html_36b62200.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5536/Chapter%203_html_36b62200.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5536/Chapter%203_html_37e40b50.gif)
Question 6:
Prove that: ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5537/Chapter%203_html_2d0bc6f8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5537/Chapter%203_html_2d0bc6f8.gif)
Answer:
It is known that
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5537/Chapter%203_html_4af3e5f6.gif)
L.H.S. = ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5537/Chapter%203_html_31d32709.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5537/Chapter%203_html_31d32709.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5537/Chapter%203_html_m23c4bf16.gif)
= tan 6x
= R.H.S.
Question 7:
Prove that: ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5538/Chapter%203_html_2ad9910c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5538/Chapter%203_html_2ad9910c.gif)
Answer:
L.H.S. = ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5538/Chapter%203_html_6ad2c8d9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5538/Chapter%203_html_6ad2c8d9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5538/Chapter%203_html_m4b0b0896.gif)
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_2ff2e2a9.gif)
Answer:
Here, x is in quadrant II.
i.e.,![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_m1445a75b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_m1445a75b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_65c8753b.gif)
Therefore,
are all positive.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_m115b87fd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_m5c0ee6e9.gif)
As x is in quadrant II, cosx is negative.
∴![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_1547ae7d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_1547ae7d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_668f3bdc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_4b90fca7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_m43d31489.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_4bb2b44a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_31b0a4b7.gif)
Thus, the respective values of
are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_m115b87fd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5061/CHAPTER%203_html_m3c699ae5.gif)
Question 9:
Find
for
, x in quadrant III
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_m50a1e778.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_6ab060e.gif)
Answer:
Here, x is in quadrant III.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_7e82a175.gif)
Therefore,
and
are negative, whereas
is positive.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_m260949f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_m3fb321a9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_m602c410.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_m3f3a448e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_25767980.gif)
Now, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_7e117052.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_7e117052.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_4db8f312.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_aa8df22.gif)
Thus, the respective values of
are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_m3c83f598.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5539/Chapter%203_html_d4dc4e6.gif)
Question 10:
Find
for
, x in quadrant II
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_m50a1e778.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_2e98af3a.gif)
Answer:
Here, x is in quadrant II.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_2ad9e074.gif)
Therefore,
, and
are all positive.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_m10d254c2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_m3fb321a9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_m134d7ec8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_59c82e04.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_345d7afd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_633affb6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_5eef340f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_1fe5b50f.gif)
Thus, the respective values of
are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_m3c83f598.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_m545668a5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5540/Chapter%203_html_m566e70be.gif)