To solve a trigonometric inequation of the type sin x ≥ a where |a| ≤ 1, we take a hill of length 2 π in the sine curve and write the solution within that hill. For the general solution, we add 2n π . For instance, to solve sinx ≥ 
, we take the hill 
over which solution is – 
. The general solution is 2n π – 
< x < 2n π + 
, n is any integer. Again to solve an inequation of the type sin x ≤ a, where |a| ≤ 1, we take a hollow of length 2 π in the sine curve. (since on a hill, sinx ≤ a is satisfied over two intervals). Similarly cos x ≥ a or cosx ≤ a, |a| ≤ 1 are solved.
(i) Solution to the inequation sin 6 x + cos 6 x < 
must be
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i) sin 6 x + cos 6 x < 
⇒ 1 – 3sin 2 x cos 2 x < 
⇒ sin 2 x cos 2 x > 
⇒ sin 2 2x > 
⇒ 
> 
⇒ 1 – cos4x > 
⇒ cos4x < 
⇒ Principal is value 4x 
General value is 
⇒ 
(ii) cos 2x + 5 cos x + 3 ≥ 0 ⇒ 2cos 2 x + 5cosx + 2 ≥ 0 ⇒ (cosx + 2)(2 cosx + 1) ≥ 0
2cosx + 1 ≥ 0 ( 
cosx + 2 > 0) ⇒ cosx ≥ – 
⇒ 
(iii) 2 sin2 
+ 
cos 2x ≥ 0
1 – cos 
+ 
⇒ 
⇒ 
⇒ 
⇒ 2x + 
⇒ 
⇒ 
⇒ 
∪ 
∪ 
in 
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