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The number of all possible triplets (a1,a2,a3)such that a1 + a2(Cos2x) +a3Sin 2 (x) = 0 for all values of x

The number of all possible triplets (a1,a2,a3)such that 
a1 + a2(Cos2x) +a3Sin2(x) = 0 for all values of x 

Grade:11

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
8 years ago
Ans:
Hello Student,
Please find answer to your question below,
a_{1} + a_{2}cos2x + a_{3}sin^{2}x = 0
a_{1} + a_{2}(1-2sin^{2}x) + a_{3}sin^{2}x = 0
a_{1} + a_{2}=sin^{2}x(2a_{2}-a_{3})
sin^{2}x = \frac{a_{1}+a_{2}}{2a_{2}-a_{3}}
Since x belong to R, we have
0\leq sin^{2}x \leq 1
0\leq \frac{a_{1}+a_{2}}{2a_{2}-a_{3}}\leq 1
0\leq a_{1}+a_{2}\leq 2a_{2} - a_{3}
a_{2}\geq a_{1} + a_{3}
a1, a2& a3belong to R with above cond. satisfied.

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