2010/043) Given that 5cos x + 12 cos y = 13 ? maximum value of 5 sin x + 12sin y

What is the maximum value of 5 sin x + 12sin y , given that 5cos x + 12 cos y = 13 ?

we start with

(5 sin x + 12sin y)^2 + (5cos x + 12 cos y)^2 = 169 + 120 cos(x-y)

or (5 sin x + 12 sin y)^2 + 169 = 169 + 120 cos(x-y)

or 5 sin x + 12 sin y = 120 cos(x-y)

5 sin x + 12 sin y = sqrt( 120 cos(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 cos x + 12 cos x = 13 so cos x = 13/17

it is possible

so maximum value of 5 sin x + 12sin y = sqrt(120)