If #sintheta + 4csctheta + 5 = 0#, what is the value of #theta#, on #[0, 2pi)#?

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Answer 1 · 2016-11-24T13:47:14

By the reciprocal identity #cscbeta = 1/sinbeta#:

#sintheta + 4/sintheta + 5 = 0#

Put on a common denominator.

#(sin^2theta + 4 + 5sintheta)/sintheta = 0#

#sin^2theta + 5sintheta + 4 = 0#

Let #t = sintheta#.

#t^2 + 5t + 4 = 0#

#(t + 4)(t + 1) = 0#

#t = -4 and -1#

#sintheta = -4 and sin theta = -1#

However, since the domain of #y = arcsinx# are #-1 ≤ x ≤ 1#, there is no real solution to #sintheta = -4#.

Hence, the only solution is #270˚# ( by the unit circle).

Hopefully this helps!