To solve 450cosΦ+225sinΦ=486.5

Question

Quick solution and explanation pls

Answer 1

Multiply and divide the left side by √[450² + 225²] = √253125 = 225√5

 

225√5 [(2/√5)cosΦ + (1/√5)sinΦ] = 486.5

 

Draw a right triangle with acute angle θ, side opposite θ of length 2 and side adjacent θ of length 1.

 

Then, sinθ = 2/√5 and cosθ = 1/√5

 

So, 225√5 [sinθcosΦ + cosθsinΦ] = 486.5

 

      225√5 sin(Φ + θ) = 486.5      θ = Sin^-1(2/√5) = 63.4°

 

      225√5 sin(Φ + 63.4°) = 486.5

 

                 sin(Φ + 63.4°) = 0.967

 

                 Φ + 63.4° = Sin^-1(0.967)  or Φ + 63.4° = 180° - Sin^-1(0.967)

 

                 Φ = 11.8° or 41.4°

 

Answer 2

450cosΦ+225sinΦ=486.5 (Multiply both sides by 2)

900cosΦ+450sinΦ=973

Since 900²/(900² + 450²) + 450²/(900² + 450²)=1, therefore we can put cosα=900/√(900² + 450²) and sinα= 450/√(900² + 450²). Then 

cosα cosΦ+sinα sinΦ=973/√(900² + 450²)

cos(α-Φ)=973/√(900² + 450²)=973/(450√5)

Φ=α-arccos(973/(450√5))+2kπ,k= 0, ±1, ±2, ±3,

 

[973/(450√5)=0.9669]