Question #e1771

1 Answer
Nghi N.
Feb 13, 2017

#tan 165 = sqrt3 - 2#

Explanation:

Call tan 165 = tan t --> tan 2t = tan 330
Trig table and unit circle give:.
#tan 330 = tan (-30 + 360) = tan (-30) = - tan 30 = - 1/sqrt3#
Use trig identity:
#tan 2t = (2tan t)/(1 - tan^2 t)#
In this case :
#-1/sqrt3 = (2tan t)/(1 - tan^2 t)#
#tan^2 t - 1 = 2sqrt3tan t#
#tan^2 t - 2sqrt3tan t - 1 = 0#
Solve this quadratic equation for tan t using improved quadratic formula (Socratic Search):
#D = d^2 = b^2 - 4ac = 12 + 4 = 16# --> #d = +- 4#.
There are 2 real roots:
#tan t = - b/(2a) +- d/(2a) = sqrt3 +- 2
tan t = sqrt3 + 2, and tan t = sqrt3 - 2
Since (165) is in Quadrant II, tan 165 is negative, then take the negative value.
tan (165) = tan t = sqrt3 - 2

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