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You have the following:
2cot2θ - 3cscθ = 0
Start by adding "3cscθ" to both sides:
2cot2θ = 3cscθ
Next, put the equation in terms of sine and cosine. This will make it easier to work with.
2cos2θ/sin2θ = 3/sinθ
Multiply both sides by sinθ to simplify:
2cos2θ/sinθ = 3
Multiply both sides by sinθ again, so the sinθ is on the other side:
2cos2θ = 3sinθ
Using the Pythagorean trigonometric identity (cos2θ + sin2θ = 1), we can put the equation entirely in terms of sine:
2(1 - sin2θ) = 3sinθ
2 - 2sin2θ = 3sinθ
Add 2sin2θ and subtract 2 from both sides, so the terms are all on one side:
0 = 2sin2θ + 3sinθ - 2
Notice how the equation looks similar to a quadratic equation. We can factor the equation just like a quadratic equation, so we get:
0 = (2sinθ - 1)(sinθ + 2)
If either piece is equal to zero, the entire equation is equal to zero. All we have to do is solve each part seperately for zero.
2sinθ - 1 = 0
2sinθ = 1
sinθ = 1/2
θ = pi/3 or 2pi/3
--
sinθ + 2 = 0
sinθ = -2
The second piece has no solution. The only two answers within the interval [0, 2pi] are pi/3 and 2pi/3.
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