If A + B + C = π , then match the items of Column-I with those of Column-II (equality must hold).
Column-I | Column-II |
(i) cos A + cos B + cos C | [A] ≤ 3 / 2 |
(ii) sin sinsin | [B] ≥ – 3 |
(iii) (cos A cos B + cos B cos C + cos C cos A) | [C] ≤ |
(iv) cos A cos B cos C | [D] ≥ – 1 |
Text Solution
Verified by Experts(i) [A]; (ii) [B]; (iii) [A]; (iv) [B]
Ans.
(i) [A], [B]
(ii) [B], [C], [D]
(iii) [A]
(iv) [B], [D]
Sol. (i) cos A + cos B + cos C beomes – 3 for A = B = π
and C = 3 π so cos A + cos B + cos C ≥ – 3 and using
y = cos A + cos B + cos C
y = 2 sin 
cos 
+ 1 – 2sin 2 
.
We get y ≤ 
.
(ii) Also, cos A + cos B + cos C
= 1 + 4 sin 
sin 
sin 
⇒ –1 ≤ sin 
sin 
sin 
≤ 
(iii) 
(cos Acos B + cos B cos C + cos C cos A) ≤ 
.
Equality holds when all of cos A, cos B, cos C are – 1.
(iv) cos A cos B cos C ≥ –1
(when all of cos A, cos B, cos C are –1).
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