Question

If a cos thetha -b sin thetha = C, then a sin thetha + b cos thetha =

Answer 1

Answer

• a sinθ + b cosθ = ? = ± √( a² + b² - c² )

Given

• a cosθ - b sinθ = c

To Find

• a sinθ + b cosθ = ?

Solution

a cosθ - b sinθ = c

Squaring on both sides , we get ,

⇒ ( a cosθ - b sinθ )² = c²

⇒ a².cos²θ + b².sin²θ - 2.acosθ.bsinθ = c² ... (1)

Now ,

a sinθ + b cosθ = ?

Squaring on both sides , we get ,

⇒ ( a sinθ + b cosθ )² = ?²

⇒ a².sin²θ + b².cos²θ + 2asinθ.bcosθ = ?² ... (2)

Now add (1) & (2) , we get ,

⇒ a².cos²θ + b².sin²θ + a².sin²θ + b².cos²θ - 2.acosθ.bsinθ + 2asinθ.bcosθ = c² + ?²

⇒ a² ( sin²θ + cos²θ ) + b² ( sin²θ + cos²θ ) = c² + ?²

⇒ a² + b² = c² + ?²

⇒ ?² = a² + b² - c²

⇒ ? = ± √( a² + b² - c² )

So , a sinθ + b cosθ = ? = ± √( a² + b² - c² )