If 0º≤x<2pi, what are all the olutions of 2 sin²Θ + 3 sin Θ + 1 = 0?

Question

a (2/3) pi, pi/2, (1/3)pi

b (7/6) pi, (3/2)p, (11/6)pi

c (5/6) pi, pi/2, pi/6

d (4/3) pi, (3/2)pi, (5/3)pi

Answer 1

Treat this like a quadratic equation by making the substitution z = sin(q)

2 sin^2q +3sin q + 1 = 0 ---&gt; 2z^2+3z+1 = 0

Factor --&gt; (2z +1)(z+1) = 0  ---&gt; z =-1 and z = -1/2

Then sin(q) =-1,-1/2  Now sin(q) is positive in first and second quadrants so look at angles &gt; pi.

sin(q) = -1 when q  = 3pi/2

sin(q) = -1/2 when q = pi+ pi/6 and q = 3pi/2+ pi/3 --&gt; q = 7pi/6, 11/6 pi

Answer is b.

Answer 2

Answer is b.

Explanation:

Let x = sin theta. Then,

2x^2 + 3x + 1 =0

(2x+1) (x+1)=0

which yields

x= -1/2 and x= -1

Both answers are acceptable.

For sin theta= -1/2, theta=7/6pi and 11/6pi

For sin theta= -1, theta= 3/2pi

Hence the answer.