How do you solve #tan^2(3x)=3# and find all exact general solutions?

1 Answer
Nghi N.
Jul 26, 2016

#x = pi/9 + (kpi)/3#
#x = (2pi)/9 + (kpi)/3#

Explanation:

#tan^2 (3x) = 3#
#tan (3x) = +- sqrt3#
There are 2 solutions. Use trig table and unit circle

a. #tan (3x) = sqrt3 = tan (pi/3)#
#3x = (pi/3) + kpi #
#x = (pi/9) + (kpi)/3#

b. #tan (3x) = -sqrt3 = tan (2pi)/3#
#3x = (2pi)/3 + kpi#
#x = (2pi)/9 + (kpi)/3#
Check by calculator:
#x = (pi/9) = 20^@# --> #3x = 60^@# --> #tan 3x = sqrt3 #-->
#tan^2 3x = 3 #. OK
#x = (2pi)/9 = 40^@# --> #3x = 120^@# --> #tan 3x = tan 120 = - sqrt3# --> #tan^2 3x = 3 #. OK

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