find all exact solutions on [0, 2pi)

Question

2sin(2t)+3sint=0

Answer 1

2sin(2t) + 3sint = 0, 0 ≤ t < 2π

2(2sintcost) + 3sint = 0

sint(4cost + 3) = 0

So, sint = 0 or cost = -0.75

sint = 0 when t = 0 or π

If cost = -0.75, then t = π - Cos^-1(0.75) or t = π + Cos^-1(0.75)

Solution set (in radians): {0, π, π - Cos^-1(0.75), π + Cos^-1(0.75)}

Answer 2

2sin(2t) + 3sint= 0

2(2sintcost) +3sint=0

4sintcost +3sint = 0

sint(4cost+3)=0

set each factor=0 and solve for sint and cost, then solve for t

sint=0

t = arcsin(0)=0

and t=pi-0=pi

t=0, pi

4cost+3=0

cost=-3/4

t= arccos(-3/4) =about 138.59 degrees

or 360-arc(-3/4) = 360-138.59 =about 221.41 degrees

t= about 138.59 and 221.41 degrees

Exact solutions are:

0, pi, arccos(-3/4), and 2pi-arccos(-3/4)

or

0, pi, pi-arcos(3/4), and pi+arccos(3/4)

either are equivalent, as arccos(3/4)=about 41.41 degrees, pi+41.41=about 221.41 degrees

and pi-41.41= about 138.59 degrees