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No. of solutions of x,where x belongs to [0,2pie] and satisfies the equation, |sin a| +|cos a | = |sin x| where a is any real number. (Plz provide solution with explanation)

No. of solutions of x,where x belongs to [0,2pie] and satisfies the equation,
|sin a| +|cos a | = |sin x| where a is any real number.
(Plz provide solution with explanation)

Grade:

1 Answers

Sandeep Pathak
askIITians Faculty 25 Points
7 years ago
Solution: 2

Consider the left hand side of the equation and find its range. To do that let’s take its square
(|sin a| +|cos a |)2 =sin2a+cos2a+|sin 2a| = 1 + |sin 2a|
Since 0 <= |sin2a| <= 1
so, 1<= (|sin a| +|cos a |)2 <= 2
Hence, we get
|sin x| >=1
which is only possible when |sin x| = 1. That happens only for x = pi/2, 3pi/2. Hence, only 2 solutions are possible.

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