If x = a cos theta + b sin theta and y = a sin theta - b cos theta - Maths - Trigonometry

Question

If x = a cos theta + b sin theta and y = a sin theta - b cos theta
prove that x square + y square = a square + b square

Ankita Agarwal · Accepted Answer

Dear Student,

x = a cosθ + b sinθ and y = a sinθ - b
cosθ
L H S = x2 + y2
= (a cosθ + b sinθ)2 + (a sinθ -b
cosθ)2
= a2cos2θ + 2ab cosθ sinθ +
b2sin2θ +
a2sin2θ - 2absinθ cosθ +
b2cos2θ
= (a2 + b2) cos2θ +
(b2+a2) sin2θ
= (a2 + b2)cos2θ +
(a2+b2) sin2θ
= (a2 + b2)(cos2θ +
sin2θ)
= (a2 + b2) [âˆµ
cos2θ + sin2θ = 1]
= R H S

Hence Proved

Regards

If x = a cos theta + b sin theta and y = a sin theta - b cos theta prove that x square + y square = a square + b square