How do you solve $tanx(cscx+2)=0$ and find all solutions in the interval $0<=x<360$ ?

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Answer 1 · 2018-04-13T17:39:35

$color(blue)(120^@,240^@)$

Identity:

$color(red)bb(cscx=1/sinx)$

Using this in given equation:

$tanx(1/sinx+2)=0$

This means either:

$tanx=0 or 1/sinx+2=0$

$tanx=0$

$x=arctan(tanx)=arctan(0)=>x=0^@, 180^@$

For:

$x=0^@ and x=180^@$

$cscx$ is undefined, so these are disregarded.

$1/cosx+2=0$

$1/cosx=-2$

$cosx=-1/2$

$x=arccos(cosx)=arccos(-1/2)=>x=120^@,240^@$

Solutions:

$color(blue)(120^@,240^@)$

How do you solve tanx(cscx+2)=0tanx(cscx+2)=0 and find all solutions in the interval 0<=x<3600<=x<360?