Question

help to solve Q......

Answer 1

a cos - b sin = c

( a cos - b sin)^2 = c^2

a^2 cos^2 + b^2 sin^2 - 2ab cos sin = c^2

a^2 cos^2 + b^2( 1 - cos^2) - 2 ab cos sin = c^2

( a^2 - b^2) cos^2 - 2ab cos sin + b^2 - c^2 = 0

( b^2 - a^2) cos^2 + 2ab sin cos +c^2 - b^2 = 0

( b^2 - a^2) cos^2 + 2ab sin cos = b^2 - c^2........(1)

NOW

( a sin + b cos)^2 = a^2 sin^2 + b^2 cos^2 + 2ab sin cos

= a^2( 1 - cos^2) + b^2 cos^2 + 2ab sin cos

= b^2 - a^2) cos^2 + 2ab sin cos + a^2

Using (1)

b^2 - c^2 + a^2

So a sin + b cos = √ b^2 - c^2 + a^2