Write -4sin(7t) + 9cos(7t) in the form of Asin(Bt+ϕ) using sum or difference formulas

Question

Write -4sin(7t) + 9cos(7t) in the form of Asin(Bt+ϕ) using sum or difference formulas I got sqrt(97)sin(7(t)-1.15257) but there homework software is telling me that: The angle phi should be in the second quadrant (not the fourth). Hint: what is the period of the tangent function?

Answer 1

If we denote by sin(ϕ)=4/√(4²+9²),  cos(ϕ)=9/√(4²+9²), then

-4sin(7t) + 9cos(7t)=

=√(4²+9²⁾)[(-4/√(4²+9²))sin(7t)+(9/√(4²+9²))cos(7t)]=

=√(16+81))[-cos(ϕ)sin(7t)+sin(ϕ)cos(7t)]=

=√97sin(7t-ϕ).

So 

-4sin(7t) + 9cos(7t)=√97sin(7t-ϕ).