Consider an equation sinx + sin y = 2 ...(1)
We know that sinx ≤ 1 and sin y ≤ 1 for all x, y so, sinx + sin y ≤ 2 for all x and y i.e. sinx + sin y = 2 if and only if sin x =1= sin y so, x = 2n π + π /2 and y = 2m π + π /2. In general, one or more of the following extreme value conditions
1) – 1 ≤ sinx ≤ 1 ⇒ |sinx | ≤ 1 and sin2x ≤ 1
2) –1 ≤ cos x ≤ 1 ⇒ |cosx| ≤ 1 and cos2x ≤ 1
3) 
≤ (a sinx + b cos x) ≤ 
⇒ |a sinx + b cos x| ≤ 
On the basis of above passage, answer the following questions:
(i) Number of roots of the equation cos7x + sin4x = 1 in the interval [0, 2 π ] is-
sin2x ≤ 1/4
(iii) The value of the k for which the equation sin x + cos (k + x) + cos (k –x) = 2 has real solutions, satisfy- cos2x ≥ 3/4
Text Solution
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Ans.
(i)
Sol. 4
(ii)
Sol. 
(iii) ,
Sol. sin2x ≤ 1/4 , cos2x ≥ ¾
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