Question

if a cos∅- bcos∅ =c then a sin ∅ + b cos∅ is equal to?​

Answer 1

Answer:

± √( a² + b² - c² )

Step-by-step explanation:

Given :

a cos ∅ - b cos ∅ = c

Squaring on both sides

⇒ ( a cos ∅ - b sin ∅ )² = c²

Using algebraic identity ( x - y )² = x² + y² - 2xy

⇒ ( a cos ∅ )² + ( b sin ∅ )² - 2absin ∅.cos ∅ = c²

⇒ a²cos² ∅ + b²sin² ∅ - 2ab. sin ∅.cos ∅ = c²

Using trigonometric identity sin² ∅ = 1 - cos² ∅ and cos² ∅ = 1 - sin² ∅

⇒ a² ( 1 - sin² ∅ ) + b² ( 1 - cos² ∅ ) - 2ab. sin ∅. cos ∅ = c²

⇒ a² - a²sin² ∅ + b² - b²cos² ∅ - 2ab. sin ∅.cos ∅ = c²

⇒ a² + b² - c² = a²sin² ∅ + b²cos² ∅ + 2ab. sin ∅.cos ∅

⇒ a²sin² ∅ + b²cos² ∅ + 2ab. sin ∅.cos ∅ = a² + b² - c²

⇒ ( asin ∅ )² + ( bcos ∅ )² + 2ab. sin ∅.cos ∅ = a² + b² - c²

Since x² + y² + 2xy = ( x + y )²

⇒ ( asin ∅ + bcos ∅ )² = a² + b² - c²

Taking square root on both sides

⇒ asin ∅ + bcos ∅ = ± √( a² + b² - c² )

Hence we found the required answer.