NCERT solution class 10 chapter 8 Introduction to Trigonometry exercise 8.4 mathematics

EXERCISE 8.4

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Q 1, Ex 8.4 - Trigonometry - Chapter 8 - Maths Class 10th - NCERT

Page No 193:

Question 1:

Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

Answer:

We know that,

 will always be positive as we are adding two positive quantities.
Therefore, 
We know that, 
However, 
Therefore, 
Also,

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Q 2, Ex 8.4 - Trigonometry - Chapter 8 - Maths Class 10th - NCERT

Question 2:

Write all the other trigonometric ratios of ∠A in terms of sec A.

Answer:

We know that,

Also, sin² A + cos² A = 1
sin² A = 1 − cos² A

tan²A + 1 = sec²A
tan²A = sec²A − 1

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Question 3:

Evaluate
(i) 
(ii) sin25° cos65° + cos25° sin65°

Answer:

(i) 


 (As sin²A + cos²A = 1)
= 1
(ii) sin25° cos65° + cos25° sin65°

= sin²25° + cos²25°
= 1 (As sin²A + cos²A = 1)

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Q 4 (i &ii), Ex 8.4 - Trigonometry - Chapter 8 - Maths Class 10th - NCERT

Q 4 (iii & iv), Ex 8.4 - Trigonometry - Chapter 8 - Maths Class 10th - NCERT

Question 4:

Choose the correct option. Justify your choice.
(i) 9 sec² A − 9 tan² A =
(A) 1
(B) 9
(C) 8
(D) 0
(ii) (1 + tan θ + sec θ) (1 + cot θ − cosec θ)
(A) 0
(B) 1
(C) 2
(D) −1
(iii) (secA + tanA) (1 − sinA) =
(A) secA
(B) sinA
(C) cosecA
(D) cosA
(iv) 
(A) sec² A
(B) −1
(C) cot² A
(D) tan² A

Answer:

(i) 9 sec²A − 9 tan²A
= 9 (sec²A − tan²A)
= 9 (1) [As sec² A − tan² A = 1]
= 9
Hence, alternative (B) is correct.
(ii)
(1 + tan θ + sec θ) (1 + cot θ − cosec θ)

Hence, alternative (C) is correct.
(iii) (secA + tanA) (1 − sinA)

= cosA
Hence, alternative (D) is correct.
(iv) 

Hence, alternative (D) is correct.

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Q 5 (i & ii), Ex 8.4 - Trigonometry - Chapter 8 - Maths Class 10th - NCERT

Q 5 (iii), Ex 8.4 - Trigonometry - Chapter 8 - Maths Class 10th - NCERT

Q 5 (vi), Ex 8.4 - Trigonometry - Chapter 8 - Maths Class 10th - NCERT

Q 5 (viii), Ex 8.4 - Trigonometry - Chapter 8 - Maths Class 10th - NCERT

Q 5 (ix), Ex 8.4 - Trigonometry - Chapter 8 - Maths Class 10th - NCERT

Q 5 (x), Ex 8.4 - Trigonometry - Chapter 8 - Maths Class 10th - NCERT

Question 5:

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

Answer:

(i) 

(ii) 

(iii) 



= secθ cosec θ +
= R.H.S.
(iv) 

= R.H.S
(v) 
Using the identity cosec² = 1 + cot²,
L.H.S = 

= cosec A + cot A
= R.H.S
(vi) 

(vii) 

(viii) 

(ix) 


Hence, L.H.S = R.H.S
(x) 

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