Trigonometric EquationHard
Question
The equation sin6 θ + cos6 θ = a has a real solution if -
Options
A.1/2 ≤ a ≤ 1
B.1/4 ≤ a ≤ 1
C.-1 ≤ a ≤ 1
D.0 ≤ a ≤ 1/2
Solution
We know a3 + b3 = (a + b)3 - 3ab (a + b)
(sin2θ + cos2θ)3 - 3 sin2θ cos2θ (sin2θ + cos2θ ) = a
1- 3sin2θ cos2θ = a
1 -
sin22θ = a
Now 0 ≤
sin22θ ≤ 
, for all
⇒ -
≤ - 
sin22θ ≤ 0, for all θ
or 1 -
≤ 1 - 
sin22θ ≤ 1 for all
≤ 1 - 
sin22θ ≤ 1 for all
⇒
≤ a ≤ 1 ⇒ 
≤ a ≤ 1
(sin2θ + cos2θ)3 - 3 sin2θ cos2θ (sin2θ + cos2θ ) = a
1- 3sin2θ cos2θ = a
1 -
sin22θ = aNow 0 ≤
sin22θ ≤ , for all⇒ -
≤ - sin22θ ≤ 0, for all θor 1 -
≤ 1 - sin22θ ≤ 1 for all ≤ 1 - sin22θ ≤ 1 for all ⇒
≤ a ≤ 1 ⇒ ≤ a ≤ 1Create a free account to view solution
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