Question

if a cos theta - b sin theta =x prove that a sin theta +b cos theta =√a2+b2-x2. pls tell fast urgenttt

Answer 1

Given:-

aCosØ - bSinØ = x ------ (1)

Let,

aSinØ + bCosØ = y -----(2)

Squaring and adding both the equations :-

(aCosØ - bSinØ)² + (aSinØ+bCosØ)² = x² + y²

a²Cos²Ø + b²Sin²Ø - 2aCosØ.bSinØ + a²Sin²Ø + b²Cos²Ø + 2aSinØ.bCosØ = x² + y²

a²Cos²Ø+a²Sin²Ø + b²Sin²Ø + b²Cos²Ø + 2ab - 2ab = x²+y²

a²(Cos²Ø+Sin²Ø) + b²(Sin²Ø+Cos²Ø) = x² + y²

a² + b² = x² + y²

y = √a²+b²-x²

Putting the value of y...

aSinØ + bCosØ = √a²+b²-x²

Hence, Proved....