Chapter 3. Trigonometric Functions

Class 11 Maths Chapter 3. Trigonometric Functions Exercise 3.1 Solutions :

1. Find the radian measures corresponding to the following degree measures:
(i)  25°      (ii)  – 47°30′      (iii) 240°      (iv) 520°

Solution:    (i) 25°

We have, 

 

(ii) 

We have,  

  

 

(iii) 240°

We have, 

 

(iv) 520°

We have,

2 . Find the degree measures corresponding to the following radian measures (Use  ).
(i)           (ii) – 4       (iii)        (iv) 

Solution:  (i)   

We have, 

 

 

Solution: (ii)  – 4

We have, 

 

   [approx]

Solution:  (ii)  

We have,

Solution:  (iv) 

We have,

 

3. A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

Solution:  The number of revolution in one minute

Now ,   one revolution

So, 6 revolution
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 (Use ).

Solution:  Here,   ;

We know that,  

 

We have, 

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.

Solution:  Here,  ,   and 

We know that , the length of minor arc of the chord

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:  For first circle : let  and  are the radius and the arc length of the circle .

    .

For second circle : let  and  are the radius and the arc length of the circle .

  

  [  ]

So, 

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:  (i) Here,  ,

We know that ,

(ii) Here,  ,

We know that ,

(iii) Here,  ,

We know that ,

Class 11 Maths Chapter 3. Trigonometric Functions Exercise 3.2 Solutions :

Find the values of other five trigonometric functions in Exercises 1 to 5.

1.If    lies in third quadrant,find the values of other five trigonometric functions .

Solution:  We have,    (i.e., third quadrant)

Since  lies in third quadrant,  is negative.

2. If lies in second quadrant.,find the values of other five trigonometric functions .

Solution:  We have,  (i.e., second quadrant.)

 

Since  lies in second quadrant,  will be negative.

Therefore,

3. If lies in third quadrant,find the values of other five trigonometric functions .

Solution:  We have,   (i.e., third quadrant)

4. If  lies in fourth quadrant,find the values of other five trigonometric functions .

Solution:    We have,   (i.e., fourth quadrant.)

5. If  lies in second quadrant,find the values of other five trigonometric functions .

Solution:  We have,  (i.e., second quadrant.)

Therefore,

Find the values of the trigonometric functions in Exercises 6 to 10.

6.         

Solution: We have,

7.        

Solution: We have,  

 

            

8.

Solution: We have,

9.  

Solution: We have, 

10.

Solution:  We have,

Class 11 Maths Chapter 3. Trigonometric Functions Exercise 3.3 Solutions :

1. Prove that:

Solution:  LHS :

RHS  Proved.

2.Prove that :

Solution:  LHS : 

  RHS  Proved.

3. Prove that :

Solution:  LHS :

  RHS    Proved.

4. Prove that :

Solution:   LHS :

  RHS Proved.

5. Find the value of:   (i) sin 75°           (ii) tan 15°

Solution:  (i) We have,

(ii) tan 15°

We have, 

6. Prove that:

Solution:  LHS :

 RHS   Proved.

7. Prove that:

Solution:  LHS:  

   RHS Proved.

8. Prove that:

Solution:  LHS :

RHS  Proved.

9. Prove that: 

Solution:  LHS :

 RHS Proved.

10. Prove that:  

Solution:

LHS :

 RHS Proved

11. Prove that:

Solution:  LHS :

   RHS Proved.

12.Prove that:  

Solution:  LHS :

  RHS  Proved.

13. Prove that :

Solution: LHS : 

 RHS Proved.

14.Prove that:  

Solution: LHS : 

 RHS Proved.

15. Prove that:  

Solution:  LHS :

RHS :

 

LHS = RHS  Proved.

16.Prove that:      

Solution: LHS :

   RHS Proved.

17.Prove that:

Solution:  LHS :

18.Prove that:

Solution: LHS :

  RHS    Proved.

19. Prove that:

Solution:    LHS :

  RHS   Proved.

20. Prove that:

Solution:  LHS :

21. Prove that: 

Solution: LHS : 

  RHS  Proved.

22. Prove that:

Solution:    We have,

 

Proved .

23. Prove that:

Solution: LHS :

  RHS Proved.

24.Prove that:   

Solution: LHS : 

 RHS

25.Prove that: 

Solution:  LHS:

 RHS Proved.

Class 11 Maths Chapter 3. Trigonometric Functions Exercise 3.4 Solutions :

Find the principal and general solutions of the following equations:
1.        2.        3.      4.

Solution:  1.                

We have,

Therefore, the  principal value is  .

We know that, 

   

So,

 

Therefore, the general value is .

Solution: 2.

We have,

Therefore, the  principal value is  .

We know that, 

   

So,

Therefore, the general value is .

Solution: 3.             

We have,

 

 

 

Therefore, the  principal value is  .

We know that, 

                

So, 

 

Therefore, the general value is  

Solution:  4.

We have,

 

 

Therefore, the  principal value is  .

We know that,

     

So, 

Therefore, the general value is  

Find the general solution for each of the following equations:
5.     6.     7.    8.      9.  

5. Solution: We have,

 

or    

Therefore, the general solution are  and  .

6. Solution:  We have,

  

or 

Therefore, the general values are   and  .

7.  Solution:  We have,       

or    

Therefore, the general values are  and  .

8. Solution:  We have,

Therefore, 

Therefore, the general values are  and  .

9.  Solution: We have,

 Or  

Therefore, the general values are  and  .

Class 11 Maths Chapter 3. Trigonometric Functions Miscellaneous Exercise Solutions :

1. Prove that: 

Solution:  LHS :

 RHS

2. Prov that :

Solution: LHS :

  RHS

3. Prove that :

Solution :  LHS : 

  RHS   Proved.

4. Prove that

Solution: LHS :

  RHS  Proved.

5.Prove that : 

Solution:  We have,

   RHS  Proved.

6. Prove that : 

6. Prove that :

Solution:  LHS :

  

 RHS Proved.

7. Prove that :

Solution:  LHS :

   RHS Proved.

8. If lies in second quadrant, find the values of  and   .

Solution:  We have,  

    (i.e., second quadrant.)

 

 

Since , 

So,     (i.e., first quadrant.)

Again,

and 

  

9.  If  in quadrant III , find the values of  and   .

Solution:

Solution:  We have,

  (i.e., third quadrant.)

So,  

Since, 

So,    (i.e., second quadrant.)

10.  If  in quadrant II, find the values of  and   .

Solution:  Since,  in quadrant II.

i.e.,  

We have, 

 

 

  [ Rationalise]

 

So,    (i.e., first quadrant .)