Exercise 4.3 - Chapter 4 - Trigonometry - 11th Business Maths Guide Samacheer Kalvi Solutions

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Exercise 4.3

Text Book Back Questions and Answers

Question 1.
Express each of the following as the sum or difference of sine or cosine:

Question 2.
Express each of the following as the product of sine and cosine
(i) sin A + sin 2A
(ii) cos 2A + cos 4A
(iii) sin 6θ – sin 2θ
(iv) cos 2θ – cos θ
Solution:

Question 3.
Prove that
(i) cos 20° cos 40° cos 80° = 1/8
(ii) tan 20° tan 40° tan 80° = √3.
Solution:

(Multiply and divide by 2)

[∵ sin(180° – θ) = sin θ]

(ii) tan 20° tan 40° tan 80°

Consider sin 20° × sin 40° sin 80°
= sin 20° sin (60° – 20°) sin (60° + 20°)
= sin 20° [sin² 60° – sin² 20°]3

Question 4.
Prove that
(i) (cos α – cos β)² + (sin α – sin β)² 

(ii) sin A sin(60° + A) sin(60° – A) = sin 3A
Solution:
(i) LHS = (cos α – cos β)² + (sin α – sin β)²

(ii) LHS = 4 sin A sin (60° + A) . sin (60° – A)
= 4 sin A {sin (60° + A) . sin (60° – A)}

= 4 sin A {sin² 60° – sin² A}
= 4 sin A {3/4 – sin² A}
= 3 sin A – 4 sin³ A
= sin 3A
= RHS

Question 5.
Prove that
(i) sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0

Solution:
Consider sin (A – B) sin C
= (sin A cos B – cos A sin B) sin C
= sin A cos B sin C – cos A sin B sin C …….. (1)
Similarly sin(B – C) sin A = sin B cos C sin A – cos B sin C sin A …….. (2)
[Replace A by B, B by C, C by A in (1)]
and sin(C – A) sin B [Replace A by B, B by C, C by A in (2)]
= sin C cos A sin B – cos C sin A sin B …….. (3)
Adding (1), (2) and (3) we get
sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0

Question 6.
Prove that

Solution:

Question 7.
Prove that cos 20° cos 40° cos 60° cos 80° 

Solution:
LHS = cos 20° cos 40° cos 60° cos 80°

[multiply and divide by 2]

Hence Proved.

Question 8.
Evaluate:
(i) cos 20° + cos 100° + cos 140°
(ii) sin 50° – sin 70° + sin 10°

Solution:
(i) LHS = (cos 20° + cos 100°) + cos 140°

= cos 40° – cos 40°
= 0
Hence Proved.

(ii) LHS = (sin 50° – sin 70°) + sin 10°

= -sin 10° + sin 10°
= 0
= RHS

Question 9.

Solution:
Given that cos A + cos B = ½

Also given that sin A + sin B = ¼

Question 10.
If sin(y + z – x), sin(z + x – y), sin(x + y – z) are in A.P, then prove that tan x, tan y and tan z are in A.P.
Solution:
In A.P. commom difference are equal, namely t₂ – t₁ = t₃ – t₂
sin(z + x – y) – sin(y + z – x) = sin(x + y – z) – sin(z + x – y)

cos z sin (x – y) = cos x sin (y – z)
cos z (sin x cos y – cos x sin y) = cos x (sin y cos z – cos y sin z)
Divide bothsides by cos x cos y cos z we get

tan x – tan y = tan y – tan z
Multiply both sides by (-1) we get,
tan y – tan x = tan z – tan y
This means tan x, tan y, and tan z are in A.P.
Hence proved.

Question 11.
If cosec A + sec A = cosec B + sec B prove that 

Solution:
Given that cosec A + sec A = cosec B + sec B

Arrange T-ratios of the sine and cosine in the separate side