Trig Question Help Really Needed

Question

Solve for x where 0 < x < pi/2.

4sqrt(3) csc x -4 - 3 csc^2 x = 0

Answer 1

-3U² +4√3 U -4 = 0 is equivalent to:

3U² -4√3 U +4 = 0 (multiply by -1)

Use the quadratic formula (easier than attempting to factor:)

U = 4√3 ± √(48 - 48)

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6

= 2√3/3

But in this case U is really csc x.

So csc x = 2√3/3.

Easier to work with sin, sin x = reciprocal of csc.

sinx = 3/2√3 = √3/2

x = arcsin (√3/2)

What angles between 0 and pi/2 (first quadrant) have a sin with a value of √3/2. Think 30-60-90 right triangle. x = 60 degrees or in radians π/3.