Sin(x+20) = Cos(3x)
Critical Thinking: What have we here?
Different ratios: Little chance of reducing to a single ratio and
argument.
We have two equations, y=Sin(x+20), y=Cos(3x). If we take Sin as
static then Cos(3x) will cut Sin(x+20) in 6 places on (0 <=x<=360). 6
solutions. Simultaneous equations. With Cos(3x) alternating 3 times
faster that Sin(x+20) expect solutions in all sectors. Notice that
Cos(x)=Cos(-x).
What to do? Convert Cos(3x) to Sin(90-3x) and equate arguments.
Explore solutions with +nk; k iamo Z
Sin (x+20) = Cos(3x) = Sin(90-3x)
Sin (x+20) = Cos(-3x) = Sin(90+3x)
x+20
4x
=
=
90-3x
70
x+20
2x
=
=
90+3x
-70
Q1:
reference angle
`
reference angle
4x =
70 + 360k
k iamo Z
x =
17.5 + 90k
k iamo Z
thus
x =
17.5, 107.5, 197.5, 287.5 on 0<=x<=360 ===>
We need 2 more solutions:
2x =
-70 + 360k
k iamo Z
x =
-35 + 180k
k iamo Z
thus
x =
325, 245
on 0<=x<=360 ===>