NCERT Solutions for Class 11 Maths Chapter 3 – Trigonometric Functions Ex 3.1
Page No 54:
Question 1:
Find the radian measures corresponding to the following degree measures:
(i) 25° (ii) – 47° 30′ (iii) 240° (iv) 520°
Answer:
(i) 25°
We know that 180° = π radian
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5497/Chapter%203_html_m5511fce1.gif)
(ii) –47° 30′
–47° 30′ =
degree [1° = 60′]
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5497/Chapter%203_html_282d3c13.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5497/Chapter%203_html_m326b3755.gif)
Since 180° = π radian
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5497/Chapter%203_html_m8dac504.gif)
(iii) 240°
We know that 180° = π radian
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5497/Chapter%203_html_m702dfd28.gif)
(iv) 520°
We know that 180° = π radian
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5497/Chapter%203_html_73b103bf.gif)
Page No 55:
Question 2:
Find the degree measures corresponding to the following radian measures
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_meb7a1f7.gif)
(i)
(ii) – 4 (iii)
(iv) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_31b827a5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_m24c81b9e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_m179f0a59.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_31b827a5.gif)
Answer:
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_m24c81b9e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_m24c81b9e.gif)
We know that π radian = 180°
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_m8135e45.gif)
(ii) – 4
We know that π radian = 180°
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_m59c35c04.gif)
(iii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_m179f0a59.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_m179f0a59.gif)
We know that π radian = 180°
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_c74946e.gif)
(iv) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_31b827a5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_31b827a5.gif)
We know that π radian = 180°
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5498/Chapter%203_html_64198d2b.gif)
Question 3:
A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?
Answer:
Number of revolutions made by the wheel in 1 minute = 360
∴Number of revolutions made by the wheel in 1 second =![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5005/CHAPTER%203_html_m1c11c552.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5005/CHAPTER%203_html_m1c11c552.gif)
In one complete revolution, the wheel turns an angle of 2π radian.
Hence, in 6 complete revolutions, it will turn an angle of 6 × 2π radian, i.e.,
12 π radian
Thus, in one second, the wheel turns an angle of 12π radian.
Question 4:
Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5499/Chapter%203_html_m7f030a7b.gif)
Answer:
We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5499/Chapter%203_html_m399eb0ee.gif)
Therefore, forr = 100 cm, l = 22 cm, we have
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5499/Chapter%203_html_m6f6a7a70.gif)
Thus, the required angle is 12°36′.
Question 5:
In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.
Answer:
Diameter of the circle = 40 cm
∴Radius (r) of the circle =![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5007/CHAPTER%203_html_7ce61241.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5007/CHAPTER%203_html_7ce61241.gif)
Let AB be a chord (length = 20 cm) of the circle.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5007/CHAPTER%203_html_m12ebff2b.jpg)
In ΔOAB, OA = OB = Radius of circle = 20 cm
Also, AB = 20 cm
Thus, ΔOAB is an equilateral triangle.
∴θ = 60° =![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5007/CHAPTER%203_html_40f07a87.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5007/CHAPTER%203_html_40f07a87.gif)
We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5007/CHAPTER%203_html_m69189415.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5007/CHAPTER%203_html_3bdee639.gif)
Thus, the length of the minor arc of the chord is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5007/CHAPTER%203_html_m7139dae3.gif)
Question 6:
If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.
Answer:
Let the radii of the two circles be
and
. Let an arc of length l subtend an angle of 60° at the centre of the circle of radius r1, while let an arc of length l subtend an angle of 75° at the centre of the circle of radius r2.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5008/CHAPTER%203_html_755a413b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5008/CHAPTER%203_html_m4a1afc13.gif)
Now, 60° =
and 75° =![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5008/CHAPTER%203_html_m3b7eabf0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5008/CHAPTER%203_html_40f07a87.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5008/CHAPTER%203_html_m3b7eabf0.gif)
We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5008/CHAPTER%203_html_31575379.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5008/CHAPTER%203_html_m38ca7cc6.gif)
Thus, the ratio of the radii is 5:4.
Question 7:
Find the angle in radian though which a pendulum swings if its length is 75 cm and the tip describes an arc of length
(i) 10 cm (ii) 15 cm (iii) 21 cm
Answer:
We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5009/CHAPTER%203_html_m69189415.gif)
It is given that r = 75 cm
(i) Here, l = 10 cm
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5009/CHAPTER%203_html_m6810e2f2.gif)
(ii) Here, l = 15 cm
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5009/CHAPTER%203_html_m30dc3f.gif)
(iii) Here, l = 21 cm
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/163/5009/CHAPTER%203_html_191556dc.gif)