SOLUTION: Solve the equation for exact solution over the interval [0,360) tan Θ csc Θ - √3 csc Θ = 0

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Question 808848: Solve the equation for exact solution over the interval [0,360)
tan Θ csc Θ - √3 csc Θ = 0
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Solve the equation for exact solution over the interval [0,360)
tan Θ csc Θ - √3 csc Θ = 0
------------
(sin/cos)(1/sin) - sqrt(3)*(1/sin) = 0
----
1/cos - (sqrt(3))/(sin)
----
(sin/cos) = sqrt(3)
tan(theta) = sqrt(3)
theta = (2/3)pi in QI
theta = (5/3)pi in QIII
==========================
Cheers,
Stan H.
==========================

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!






So



is equivalent to



Add



to both sides:



Cross multiply:



Square both sides



Use the Pythagorean identity







From which we get, using the unit circle, that



Which then implies that

But if sine and cosine are opposite signs, the original equation fails. Hence, the solution set is confined to Quadrants I and III where the signs of sine and cosine are the same. And the solution set is therefore:



John

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