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^@)#