COMBINE TRIGONOMETRIC FUNCTIONS SOP
When we solve the differential equation
ay 00 + by 0 + cy = 0,
(1)
we will have the general solution
√
√
4ac − b2
4ac − b2 − −b
t
2a
c1 cos
y=e
t + c2 sin
t
2a
2a
if b2 < 4ac. This notes is aimed to combine the trigonometric
functions in the
√
4ac − b2
. We intend to
parenthesis. For short we denote A = c1 , B = c2 , ω =
2a
compute
A cos ωt + B sin ωt.
√
• Step 1. Factor out A2 + B 2 artificially. Compute
A cos ωt + B sin ωt
p
B
A
cos ωt + √
sin ωt ,
(2)
= A2 + B 2 √
A2 + B 2
A2 + B 2
A
B
so we can treat √
= cos δ, √
= sin δ for some δ by putting
2
2
2
A +B
A + B2
A, B on a right triangle.
• Step 2. Use trigonometric identity cos(a − b) = cos a cos b − sin a sin b. By
the trigonometric identity
cos δ cos ωt + sin δ sin ωt = cos(ωt − δ).
Hence (2) can be rewritten as
where R =
p
A cos ωt + B sin ωt = R · cos(ωt − δ),
B
A2 + B 2 , δ = tan−1 .
A
1