How do you solve $sinx+4cscx+5=0$ and find all solutions in the interval $0<=x<360$ ?

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Answer 1 · 2016-10-03T02:04:58

$x = 270°$

Given:
$sin(x) + 4csc(x) + 5 = 0; 0° <= x < 360°$

Multiply both sides by sin(x):

$sin²(x) + 5sin(x) + 4 = 0; 0° <= x < 360°$

Factor:

$(sin(x) + 4)(sin(x) + 1) = 0; 0° <= x < 360°$

$sin(x) = - 4$ and $sin(x) = -1; 0° <= x < 360°$

We must discard the $sin(x) = -4$ answer, because the sine function only returns values between -1 and 1 inclusive.

That leaves us with:

$sin(x) = -1; 0° <= x < 360°$

Take the inverse sine of both sides:

$x = sin^-1(-1); 0° <= x < 360°$

$x = 270°$

How do you solve sinx+4cscx+5=0sinx+4cscx+5=0 and find all solutions in the interval 0<=x<3600<=x<360?