Chapter 8. Introduction of Trigonometry

Class 10 Maths Chapter 8. Introduction of Trigonometry Multiple Choice Questions and Solutions :

Question : If , then the value of  is :  [SEBA 2014]

(a)  30°     (b) 45°    (c)  60°    (d) 90°

Solution : (a)  30°

[We have,      ]

Question : The value of  is :   [SEBA 2015]

(a)    (b)    (c)    (d)

Solution :  (d)

[ We have,    ]

Question : The value of is : [SEBA 2016]

(a)  – 1         (b)  0    (c)  1     (d)  2

Solution : (c)  1

[W e have,

  ]

Question : The value of  is : [SEBA 2017]

(a)       (b)   (c)     (d) 

Solution :  (d) 

[ We have,    ]

Question : The value of  is : [SEBA 2018]

(a)  92    (b)  3     (c)  18    (d)  9

Solution : (d) 9

 [ We know that ,

We have,

 ]

Question : Which of the following is true ? [SEBA 2019]

(a) The value of  is always less than 1 . 

(b) The value of is always greater than 1 .

(c) for some angle  . 

(d)  for som angle  .

Solution : (d)  for som angle  .

[(a) The value of  is always less than 1.

This statement is not true. The tangent function can have values less than 1, equal to 1, or greater than 1, depending on the angle A.

(b) The value of is always greater than 1.

This statement is not true. The cotangent function can have values less than 1, equal to 1, or greater than 1, depending on the angle A.

(c) for some angle θ.

This statement is not true. The sine function always has values between -1 and 1. The value is outside this range.

(d) for some angle θ.

This statement is true. The secant function is the reciprocal of the cosine function, and it can take values greater than 1. If , then , and this is possible for some angle θ.

So, the correct answer is (d)  for some angle θ. ]

Question:    [SEBA 2020]

(a) 0        (b)  1     (c)  8     (d)  16

Solution :  (c)  8

 [ We have ,

   ]

Question :  

(a)         (b)           (c)         (d)                                

Solution:  (b)   .                     

[ We have , ]

Question : 

(a) tan90°       (b) 1            (c)  sin45°           (d)  0 

Solution:   (d)  0   .

[ We have , ]

Question:   The value of cosec31° – sec59° is :

  (a)  1     (b)  – 1         (c)  0        (d) 2

Solution:    (c)  0   .

[ We have ,

  ]

Question :     

  (a)   – 77        (b)  1       (c)  77       (d)  0

Solution:  (a)   – 77                

  [ We have ,    ]

Question :  If  , then the value of   is  :

  (a)           (b)          (c)          (d) 

Solution: (a)        

[ We have , 

and  

   ]

Question : If    , then the value of  is

(a) 45°            (b) 30°         (c) 60°           (d) 90°

Solution:  (d)   90°   .

  [ We have ,  

   

  

      ] 

Question :   is equal to

(a)  – 1     (b) 1    (c)  0     (d)  

Solution:  (a)  – 1

[  We have ,  ]

Question : If    is right angled at C , then the value of is :

(a)        (b) 0      (c) 1    (d) – 1

Solution: (b) 0 

[ Given ,  

Now,       ]

Question : If   , then the value of is equal to :

(a)        (b)            (c)       (d) 

Solution:  (c)                          

[ Given , 

   ]

Question : If   , then value of   is equal to –

   (a) 0         (b)           (c)          (d)  

Solution:   (a) 0      .

 [ We have , 

 

 

 

  ]

Question :  If   , then  equals :

(a)       (b)      (c)    (d)  

Solution:   (a)

[ Given , 

and   ]

Question : If   and , then the value of  is :

(a) 45°      (b) 60°      (c)  90°     (d)  30°

Solution: (d)  30°  

[ We have ,

 

  ]

Question : The value of    is :

   (a) 1       (b)  0        (c)          (d)  

Solution:    (b)  0

  [ We have ,  

    ]

Question : If  and  , then the value of  is :

(a)  – 1     (b)  1      (c)  0     (d)  2

Solution:  (b) 0

[ We have,

  ]

Question : Given that   and , then the value of    is :

  (a) 1         (b) 2      (c) 3        (d) 

Solution:  (a) 1                          

 [ We have ,

and

    ] 

Question :  Which of the following statement  is true ?

(a)    for all values of   .

(b)  The value of  increases as  increase.         

(c)  The value of  increases as  increase.          

(d)   is not defined for  . 

Solution: (b)  The value of  increases as  increase.          

Question : The  value of   is equal to :

(a)            (b)          (c)          (d)   

Solution:  (c)                     

[ We have ,  

  ]

Question : If  , where  is an acute angle , then the measure of  is :

  (a) 35°       (b) 37°      (c) 39°      (d) 21°

Solution:  (c) 39°                            

 [ We have , 

  

  ]

Question :  The value of   is :

  (a)  0        (b)  2           (c)  – 2          (d)   1

Solution:   (c)  – 2                            

 [ We have , 

  ]

Question :  If   , then the value of   is equal to –

  (a)  3          (b)  5          (c)  6            (d)  8

Solution:  (d)  8

[ We have , 

      ]

Question :  Which of the following statement  is true ?

a) The value of    is always less than 1.   

(b) The value of   is always greater than 1.

(c)    for some angle  .                        

(d)   for some angle  .  

Solution:  (d)  for some angle  .      

 Class 10 Introduction of Trigonometry Fill in the blanks

Q1.  If   , then the value of  is    . 

Solution:   0

[ We have ,    

 

   ]

Question : If   , then the value of   is   .

Solution:  0            

[ We have,   ]

Question :  If   , then the value of  is equal to   .

Solution:       .

 [ We have , 

and

  ]

Q3. If    , then the value of the expression  is  .

Solution:  1 

[  We have ,   

  

 

   

 

      ]

Q5. If   , then the value of  is   .

Solution:      .

 [ We have ,  

 

      ]

Q6.    is equal to  .

Solution:    – 1

 [  We have ,    ]

Q7. If   , then the value of   is equal to   .

Solution:    1

 [ We have ,

 

  

   

      ]

Q8. If  , then the value of  is equal to  .

Solution:    .

 [  Given ,

Q9. If   and   , then the value of  is   .

Solution:  0

 [ We have ,

   ]

Question : If θ=30° , then  is equal to  .

Solution:       

[  We have ,   ]

Q10. If  , then  equals  

Solution:     .

 [ Given , 

  ]

Q11. The value of   is . 

Solution:  

[ We have , 

]

Q12. The value of   is   .

Solution:    1

[ We have ,

   ]

Q12. If  A and 3A – 30° are acute angles such that  , then the value of   is equal to  .

Solution:        .

 [ We have ,  

  

  

  

      ]

Class 10 Maths Chapter 8. Introduction of Trigonometry Answers following the Questions  :  

Question :   Evaluate  :   .

Solution:  We have,

Question:  Evaluate :  

Solution:  We have , 

Question :  If  , find the value of  .

Solution:  Given ,  

 [ Use ,    ]

We have ,

So,

Question :  If  , find  .

Solution:   We have, 

Question :  Write the expression in simplest form : 

Solution:  We have ,

       

Class 10 Maths Chapter 8. Introduction of Trigonometry 

    SECTION = B  [2 Marks]

Question : Express the trigonometric ratios  and in terms of  .

Solution : We have ,

 

And 

Question : 

Solution : LHS :  

Question : Evaluate :

Solution : We have,

 

 

 

Question :  If where  is an acute angle , find the value of .

Solution : We have,

 

Since  and  are both acute  angles .

  

 

 

 

Therefore, the value A is 36° .

Question :  Prove that :

Solution:  L.H.S.  :  

   R.H.S.  Proved.

Question : If  , find  and  .

Solution:  Given , 

Let ,  

  and

In  , we have

 

=  

  Therefore , 

 and 

Question :  If  and ,find and .

Solution :  We have ,

 

 

And

 

 

 

Putting  in equation , we have

 

 

 

Therefore , the value of A and B are 45° and 15° respectively .

Question : If  , and  , then find and  .

Solution:  We have, 

And

 

 

Putting  in (i) , we get

 

 

Therefore, the value of   and  .

 

Question :  If  , then find the value of  .

Solution : We have,

Since  and are both acute angles

So,

 

 

 

Therefore, the value of  is 14° .

Question :  Prove that  :    [SEBA 2016 , 19]

Solution:   L.H.S. :  

  R.H.S.    Proved.

Question :  If  , find the value of  .    [SEBA 2017 ]

Solution:  Given ,

We have ,

  and  

Now ,  

 

Question :  Evaluate :   [SEBA 2020]

Solution:  We have ,

   [ ]

Question :  Prove that :     [SEBA 2018]

Solution:  L.H.S.  : 

R.H.S.

Question :  If A , B and C are interior angles of a triangle ABC , then show that

Solution : Since, A , B and C are interior angles of a triangle ABC respectively .

Question :  Prove that :     [SEBA 15 ]

Solution:  L.H.S.  : 

 

           R.H.S     Proved.

Question :   Prove that :   

Solution:   L.H.S. :  

  R.H.S.

Question : Prove that : 

Solution:  LHS :

  RHS    Proved

Question :  Prove that :      [SEBA 2017 ]

Solution:  L.H.S.  : 

  R.H.S.

Question :  In a right triangle ABC ,right-angled at B, if ,then verify that  [SEBA 2018]

Solution:  Given ,

 

   (say)

     

In , we have

 

   Therefore ,

and

     Verified .

Question :   Show that  : [SEBA 2016,19]

Solution : We have , 

Question : 

Solution : LHS :

  RHS                   

Question :   Prove that :   [SEBA 2020]

Solution:    L.H.S :  

     R.H.S