How Do I Answer this Trig Question

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

Use

 

sin(x + y) = sinx cosy + cosx siny

 

Multiply and divide by √[(-4)² + (9)²] = √97

 

√97 [sin7t (-4/√97) + cos7t (9/√97)] = A sin(7t + φ)

 

-4/√97 = cosφ

9/√97 = sinφ

 

This puts φ in quadrant II.

 

φ = π - arctan(9/4) ≅ (π - 1.15257) rad = 1.98902 rad

 

A = √97

 

f(t) = √97 sin(7t + 1.98902)