Solve for x in sec^2(x)+5(1-tanx)=0 for 0<=x<=360?

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Answer 1 · 2018-02-24T22:44:16

See below.

Identity:

#color(red)bb(sec^2x=1+tan^2x)#

Using this identity:

#1+tan^2x+5(1-tanx)=0#

Multiply bracket and simplify:

#tan^2x+6-5tanx=0#

#tan^2x-5tanx+6=0#

Let #color(white)(88)u=tanx#

Then:

#u^2-5u+6=0#

Factor:

#(u-3)(u-2)=0=>u=3 and u=2#

But: #color(white)(88)u=tanx#

So:

#tanx=3 and tanx = 2#

#x=arctan(tanx)=arctan(3)=>color(blue)(x=arctan(3), 180^@+arctan(3))#

#~~color(blue)(71.57^@ and 251.57^@)#

#x=arctan(tanx)=arctan(2)=>color(blue)(x=arctan(2), 180^@+arctan(2))#

#~~color(blue)(63.43^@ and 243.43^@)#