If $(5 s i n x + 12 s i n y)^{2} + (5 c o s x + 12 c o s y)^{2} = 169 + 120 c o s (x - y)$ then the maximum value of $5 sin x + 12 sin y$ is

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Solution

$(5 s i n x + 12 s i n y)^{2} + (5 c o s x + 12 c o s y)^{2} = 169 + 120 c o s (x - y)$
or $(5 s i n x + 12 s i n y)^{2} + 169 = 169 + 120 c o s (x - y)$
or $5 s i n x + 12 s i n y = 120 c o s (x - y)$
$5 s i n x + 12 s i n y = \sqrt{120 c o s (x - y)}$
this is maximum when cos(x-y) is maximum
theoritically cos(x-y) is maximum $= 1$ when $x = y$
but is is possible under given case that is x =y satisfies
then we get
$5 c o s x + 12 c o s x = 13 s o c o s x = 13 / 17$
it is possible
so maximum value of $5 s i n x + 12 s i n y = \sqrt{120}$

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