Write 6sint8cost in the form AsinBt using sum or difference

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

Solution

6sint - 8*cost

=(6²+ 8²)* [ (6sint - 8cost) / (6²+ 8²))

=100*[ (6/100)*sint - (8/100)*cost]

[let cos@ = 6/100, sin@ = - 8/100,

tan@ = sin@/cos@ = -8/6, @ = tan^-1(-8/6) = - 53.1 degree]

=100* ( sint*cos@ + cost*sin@)

= 100* ( sin(t+@)

on comparing it with given eq, Asin(Bt + @)

A = 100, B = 1, @ = -53.1 degree