find the values of the constants A, R and α

Question

6cos^2 θ -8 sin θ cos θ = A + R cos (2θ + α)

Answer 1

6cos²θ - 8sinθcosθ = 3[2cos²θ - 1] + 3 - 4[2sinθcosθ]

 

                             = 3cos(2θ) - 4sin(2θ) + 3

 

Multiply and divide by √[3²+(-4)²] = 5 to obtain:

 

   5[(3/5)cos(2θ) - (4/5)sin(2θ) + 3/5]  

 

Draw a right triangle with acute angle α, side opposite α of length 4, side adjacent to α of length 3 and hypotenuse 5.  Then cosα = 3/5 and sinα = 4/5.

 

So, we now have:  5[cos(2θ)cosα - sin(2θ)sinα] + 3

 

                         = 3 + 5cos(2θ + α)

 

A = 3, R = 5, and α = Cos^-1(3/5) = 0.927