Class 11 Maths Chapter 3 Trigonometric Functions

Exercise 3.1

 Question 1.
Find the radian measures corresponding to the following degree measures:
(i) 25°
(ii) -47°30′
(iii) 240°
(iv) 520°
Solution.
We have, 180° = π Radians

 Question 2.
Find the degree measures corresponding to the following radian measures 
(i) 
(ii) -4
(iii) 
(iv) 
Solution.
We have π Radians = 180°

 Question 3.
A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second ?
Solution.
Number of revolutions made by wheel in one minute = 360
As we know that, 1 Revolution = 27 π Radians
∴ 360 Revolutions = 720 π Radians
∴ In 1 minute wheel can make = 720 π Radians
⇒ In 60 seconds wheel can make = 720 π Radians
⇒ In 1 second wheel can make

 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 
Solution.
Let O be the centre and AB be the arc length of the circle.

 Question 5.
In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of Xhe chord.
Solution.
Let AB be the minor arc of the chord.
AB = 20 cm, OA = OB = 20 cm

 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.
Solution.
Let r1 r2 and θ1, θ2 be the radii and angles subtended at the centre of two circles respectively.

 Question 7.
Find the angle in radian through 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
Solution.

 

Exercise 3.2

Find the values of other five trigonometric functions in Exercises 1 to 5.
 Question 1.
, x lies in third quadrant.
Solution.

Question 2., x lies in second quadrant.

Solution.

 Question 3.
, xlies in third quadrant.
Solution.

 Question 4.
, x lies in fourth quadrant.
Solution.

 Question 5.
, x lies in second quadrant.
Solution.

Find the values of the trigonometric functions in Exercises 6 to 10.
 Question 6.
sin 765°
Solution.

 Question 7.
cosec (-1410°)
Solution.

 Question 8.

Solution.

 Question 9.

Solution.

 Question 10.

Solution.

Exercise 3.3

 Question 1.
Prove that: 
Solution.
L.H.S. = 

 Question 2.

Solution.
L.H.S. = 

 Question 3.

Solution.
L.H.S. = 

 Question 4.

Solution.
L.H.S. = 

 Question 5.
Find the value of:
(i) sin 75°
(ii) tan 15°
Solution.
(i) sin (75°) = sin (30° + 45°)

(ii) tan 15° = tan (45° – 30°)

Prove the following:
Question 6.

Solution.
We have,

 Question 7.

Solution.
We have,

 Question 8.

Solution.
We have,

 Question 9.

Solution.
We have,

Question 10.
sin(n +1 )x sin(n + 2)x + cos(n +1 )x cos(n + 2)x = cosx
Solution.
We have,

 Question 11.

Solution.
We have,

Question 12.
sin26x – sin24x= sin2x sin10x
Solution.

Question 13.
cos22x – cos26x = sin 4x sin 8x
Solution.

 Question 14.
sin2x + 2 sin 4x + sin 6x = 4 cos2 x sin 4x
Solution.
We have,

 Question 15.
cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x)
Solution.

 Question 16.

Solution.
We have,

 Question 17.

Solution.
We have,

 Question 18.

Solution.

 Question 19.

Solution.

Question 20.

Solution.

 Question 21.

Solution.

 Question 22.
cot x cot 2x – cot 2x cot 3x – cot3x cotx = 1
Solution.
We know that 3x = 2x + x.
Therefore,

 Question 23.

Solution.

 Question 24.
cos 4x = 1 – 8 sin2x cos2x
Solution.

Question 25.
cos 6x = 32 cos6 x – 48 cos4x + 18 cos2 x -1
Solution.

Ex 3.4

Find the principal and general solutions of the following equations:
Question 1.

Solution.

 Question 2.
sec x = 2
Solution.

 Question 3.

Solution.

Question 4.
cosec x = -2
Solution.

Find the general solution for each of the following equations:
Question 5.
cos 4x = cos 2x
Solution.

 Question 6.
cos 3x + cos x – cos 2x=0
Solution.

 Question 7.
sin 2 x + cos x = 0
Solution.

 Question 8.
sec22x = 1 – tan 2x
Solution.

 Question 9.
sin x + sin 3x + sin 5x = 0
Solution.