Question

if acostheta - bsintheta =c prove that (asintheta + bcostheta)= +-✓a^2+b^2+c^2​

Answer 1

Answer:

Instead of theta I had used 'A'.

Step-by-step explanation:

acosA -bsinA= c

squaring both sides

a²cos²A +b²sin²A-2abcosAsinA=c ________(1)

a²sin²A +b²cos²A+ 2absinAcosA = (asinA+bcosA)² _____(2)

adding 1 and 2

a²cos²A + b²sin²A-2abcosAsinA + a²sin²A +b²cos²A+ 2absinAcosA = c+ (asinA+bcosA)²

(a²+b²)cos²A +(a²+b²)sin²A = c+ (asinA+bcosA)²

(a²+b²)(cos²A+sin²A)-c = (asinA+bcosA)²

We know that sin²A+cos²A=1

a²+b²-c = (asinA+bcosA)²

±√(a²+b²-c) = asinA+bcosA

asinA+bcosA = ±√(a²+b²-c)

Hence proved.