Write −6sin(8t)+5cos(8t) in the form Asin(Bt+ϕ) using sum or difference formulas.

Question

-6sin(8t)+5cos(8t)=

Please provide all steps! Thank You!

Answer 1

Let A=√((-6)²+5²)=√61

Draw a right triangle, horizontal leg from the origin to x=-6, vertical leg from (-6,0) to (-6,5). The angle Θ, in the second quadrant, from the positive x-axis to the hypotenuse, is our guy.

cosΘ=-6/A, sinΘ=5/A. Our expression now reads

A(cosΘsin8t+sinΘcos8t)=Asin(8t+Θ)