What is cscθ+2=0 Find all solutions of the equation in the interval 0,2π. answer must be in radians in terms of pi (π)

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Borys Shumyatskiy eNotes educator| Certified Educator

The original equation is csc(t) + 2 = 0.

csc(t) = 1/sin(t) so the equation can be rewritten as

1/sin(t) + 2 = 0, or sin(t) = -1/2.

Solutions in the interval (0,2π) are the angles at which unit circle intersects with the line y=-1/2. There are two such angles, let's find them.

We know that sin(30°) = 1/2, so sin(-30°) = -1/2. 30° is π/6, -30° is -π/6.
As sin(x+2π) = sin(x) we obtain sin(-π/6+2π) = sin(-π/6), sin(11π/6) = -1/2.

Also sin(π-x) = sin(x) and therefore sin(π-(-π/6)) = sin(-π/6), sin(7π/6) = -1/2.

The answer: t1=7π/6, t2=11π/6.