22. Chapter 22. Introduction to Trigonometry NIOS Class 10 Mathematics Textbook Solutions

 Chapter 22. Introduction to Trigonometry NIOS Class 10 Mathematics Textbook Solutions for Exam Preparation

Chapter 22. INTRODUCTION TO TRIGONOMETRY

CHECK YOUR PROGRESS 22.1

1. In each of the following figures, ΔABC is a right triangle, right angled at B. Find all the trigonometric ratios of θ .

Fig. 22.11

Solution: (i) Here,

  

   

  

  

and  

(ii) Here,  

  

   

 

  

 

 and 

(iii) Here,  

 

 

 

 

  

 and  

(iv)  Here,  

  

 

 

 

 

 and 

2. In ΔABC, ∠ B = 90°, BC = 5cm , AB = 4cm, and  , find the value of sin A, cos A, and tan A.

Solution: Here, ∠ B = 90°, BC = 5cm, AB = 4cm, and  

Photo

  

 

3. In ΔABC right angled at B, if AB = 40 cm, BC = 9 cm and AC = 41 cm, find the values of sinC, cotC , cosA and cotA.

Solution: Here, AB = 40 cm, BC = 9 cm and AC = 41 cm

 

 

 

 

4. In ΔABC, ∠ B = 90°. If AB = BC = 2cm and   cm, find the value of secC , cosecC , and cotC.

Solution: Here, ∠ B = 90°. If AB = BC = 2cm and   cm

 

 

5. In Fig. 22.12, ΔABC is right angled at A. Which of the following is true?

 (i)   (ii)   (iii)    (iv)

Answer:  (iv)  

[ Here, AB = 5cm , C = 12 cm , BC = 13 cm

We have,  

6. In Fig. 22.13, AC = b, BC = a and AB = c .Which of the following is true?

(i)    (ii)    (iii)    (iv)

Answer:  (ii)     

[ Here,  

 We have,    ]

CHECK YOUR PROGRESS 22.2

1. In right ΔABC, right angled at B, AC = 10 cm, and AB = 6 cm. Find the values of sinC , cosC and tanC.

Solution: Here,

In ΔABC , We have  

 

  

Now,       ,     and  

2. In ΔABC, ∠ C = 90°, BC = 24 cm and AC = 7 cm. Find the values of sinA , cosecA and cotA .

Solution: Given, ∠ C = 90°, BC = 24 cm and AC = 7 cm

In ΔABC,    

 

Now,     ,      and  

3. In ΔPQR, ∠ Q = 90°, PR =  cm and QR = 10cm. Find the values of secP, cotP and cosecP.

Solution: Here,

In ΔPQR, we have   

Now,      ,   and 

4. In ΔPQR, ∠ Q = 90°, PQ =  cm and QR = 1 cm. Find the values of tan R, cosec R, sin P and sec P.

Solution: Here,

In ΔPQR, we have  

 

Now,     ,    ,    and  

5. In ΔABC, ∠ B = 90°, AC = 25 cm, AB = 7 cm and ∠ ACB = θ. Find the values of cotθ, sinθ, secθ and tanθ.

Solution: Given,

In ΔABC, we have

 

Now,    ,   ,     and 

6. In right ΔPQR, right-angled at Q, PQ = 5 cm and PR = 7 cm. Find the values of sinP, cosP, sinR and cosR. Find the value of sinP – cosR.

Solution: Given,  

In ΔPQR , we have

  

 

 Now,   ,    ,   and  

7. ΔDEF is a right triangle at E in Fig. 22.18. If DE = 5 cm and EF = 12 cm, which of the following is true?

(i)      (ii)    (iii)    (iv) 

Answer: (iii)

[ Here, DE = 5 cm and EF = 12 cm

In ΔDEF ,   

  Now,    ]

CHECK YOUR PROGRESS 22.3

1. If  , find the values of cosθ and tanθ.

Solution: Given,   

  

,  

In we have

  

 

 

Now,        and 

2. If  , find the values of sinθ and cosθ.

Solution: Given,  

  

 

In ,we have

 

 

 Now ,   and  

3. If  , find the values of sinA and tanA.

Solution: Given, 

   

 ,  

In,we have

 

Now,    and  

4. If  , find the values of cotθ and cosecθ.

Solution: Given,   

  

   

In, we have

 

and  

5. If cosθ =45 , evaluate cosθcotθ1-sec²θ  .

Solution: Given,

  

In We have,

  

We have,      ,      and  

Now,  

6. If  , find the value of  .

Solution:  Given,

    

 

 

In, we have

 

   ,     and  

Now,   

7. If , then show that.

Solution:  Given,   

  

 

InWe have,

 

   and  

 

 

     Proved.

8. ΔABC is a right triangle with ∠C = 90° . If  , find the values of sinB and tanB .

Solution: Given,   

  

 

In,we have

  

 

Now,     and    

9. If    and   , then show that  .

Solution: Here,   and  

LHS:  

   RHS   Proved.

10. If  , show that  .

[Hint: Find the values of tan A, sin A and sec A and substitute]

Solution: Given,     

  

     and  

In ΔABC , we have,  

     ,      and  

LHS:   

 

RHS:   

LHS = RHS   Proved.

11. In Fig. 22.24, ΔABC is right-angled at vertex B. If AB = c, BC = a and CA = b, which of the following is true?

(i)             (ii)

(iii)           (iv)  

Answer:  (iii)  

[ In given figure,

Now,    ]

CHECK YOUR PROGRESS 22.4

1. If    and  , find the values of cotθ and secθ .

Solution: Here,    and  

  

2. If  and  , find the value of    .

Solution: Here,  and tanθ=3

 

  

3. In a right angled ΔABC, right angled at C,  . Find the value of  .

Solution: Here,  

Now,  

 

4. If cosec A = 2, find the value of sin A and tan A.

Solution: Here,

 

 

 

5. In a right angled ΔABC, right angled at B,  , find the value of .

Solution:  Here,

Now,  

 

 

CHECK YOUR PROGRESS 22.5

Prove each of the following identities:

1. 

Solution: LHS:  

 RHS .  Proved

2.  

Solution:  LHS:  

3.

Solution:  LHS:   

 RHS   Proved.

4.   

Solution: LHS:  

 RHS  Proved. 

5.  

Solution:  L.H.S. :  

    R.H.S.    Proved.               

6.

Solution: LHS :   

    RHS   Proved.

7.   

Solution: LHS:

 

 

8.  

Solution:  LHS:  

 

 RHS  Proved.

9.  

Solution: LHS:  

 

 

 

 RHS   Proved.

10.  

Solution:  LHS:   

 

    RHS . Proved.

 

11.  

Solution: LHS:

12.  

Solution: LHS:   

 

 

  

  

      RHS  Proved.

13.  

Solution: R.H.S :   

 

      R.H.S  Proved. 

14.  

Solution:  L.H.S :   

  

   R.H.S.  Proved.

15.

 Solution:   LHS :   

 

 

   RHS  Proved

16. If  , then show that .

Solution: We have,  

  

 

  Proved .

Select the correct alternative from the four given in each of the following questions (17 - 20):

17.  is equal to

(i) 0      (ii) 2      (iii) 1      (iv)  

Answer:  (iii)  1

[ We have,  

    ]

18.   is equal to:

(i) 1     (ii)      (iii) 0    (iv)  

Answer:  (ii)  

[ We have, 

  ]

19.  is equal to

       (i)  0      (ii)  1        (iii)      (iv)   

Answer:  (i)  0

[ We have,  

    ]

20.  is equal to

(i)  2      (ii)   1        (iii)  0       (iv)  

Answer:  (iii)  0

[ We have,  

  ]

CHECK YOUR PROGRESS 22.6

1. Show that:    (i)         (ii)        (iii)   

Solution:   (i) LHS:   RHS      

(ii) LHS:  

RHS     

(iii) LHS:   

 RHS

2. Evaluate each of the following:

(i)        (ii)              (iii)          (iv)          (v)      (vi)        (vii)         (viii)  

Solution: (i)   

(ii)    

(iii)     

(iv) 

 

(v)

 

(vi) 

 

(vii)

 

   

(viii)   

  

3. Evaluate each of the following:

(i)  

(ii)  

Solution: (i)  

 

(ii)  

 

4. Prove that :

(i)

(ii)  

(iii)  

(iv)  

(v)  

(vi)   

(vii)   

Solution:   (i) LHS:  

(ii)   LHS:

 

 RHS Proved.

(iii)     

 

 

  RHS  Proved.

(iv)   LHS :

RHS:   

 

 

LHS = RHS Proved.

(v)   LHS:   

(vi)   LHS:  

 

 

 RHS Proved.

(vii)  LHS:

 

 

 RHS  Proved.

5. Show that  

Solution:  LHS:   

 

RHS  Proved.

6. If  where A and B are acute angles, prove that A + B = 90°.

Solution:  We have,  

   Proved.

7. In a , Prove that

(i)           (ii) 

Solution: (i)  Let, A ,B  and C are the interior angle of the triangle ABC respectively .

We have,  

 

 

     Proved.

(ii) Let, A ,B  and C are the interior angle of the triangle ABC respectively .

We have,

 

   Proved.

8. Express  in terms of trigonometric ratios of angles between 0° and 45°.

Solution: We have,  

9. Express   in terms of trigonometric ratios of angles between 0° and 45°.

Solution: We have,  

10. Express  in terms of trigonometric ratios of angles between 0° and 45° .

Solution:  We have, 

Select the correct alternative for each of the following questions (11-12):

11. The value of    Is :

(i)   – 1     (ii)          (iii)          (iv)  1

Answer:   (iii)      .

[ We have ,   

     ]

12. If  , where  is an acute angle, then θ is

(i) 54°      (ii) 18°     (iii) 21°     (iv) 27°

Answer:  (iv)  27° 

[ We have,

 

 

   ]

TERMINAL EXERCISE

1. If  , find the values of cos A and tan A.

Solution: We have,  

 

 

  

2. If   , find the values of cosec A and sec A.

Solution: Here,   

 

 

And  

 

3. If  , find the value of sin θ + cos θ.

Solution: We have,  

 

 

 

 

4. If  , find the values of sin θ and tan θ.

Solution: We have,  

5. If  , find the value of  

Solution: We have,   

 

Now,   

6. If  , find the value of   .

Solution: We have,  

 

Now,  

7. If  , find the value of  .

Solution:  We have,  

Now,  

 

 

Prove each of the following identities (8 –20):

8.  

Solution:  LHS :

   RHS   Proved.

---

Posted 1 month ago