How do you use the double angle or half angle formulas to simplify tan 300?

1 Answer
Nghi N.
Oct 21, 2015

Find tan 300.

Ans: #sqrt3#

Explanation:

#tan 300 = sin 300/(cos 300)#. First find sin 300 and cos 300.
cos 600 = cos (240 + 360) = cos 240 = cos (60 + 180) =
= - cos 60 = #- 1/2#.
Apply the 2 trig identities:
#cos 2x = 1 - 2sin^2 x# and #cos 2x = 2cos^2 x - 1#
#cos 600 = - 1/2 = 2cos^2 (300) - 1#
#2cos^2 (300) = 1 - 1/2 = 1/2# --># cos^2 (300) = 1/4.#
#cos (300) = +- 1/2#. Since cos 300 is negative, then
#cos 300 = -1/2#
#cos (600) = -1/2 = 1 - 2sin^2 (300)#
#2sin^2 (300) = 1 + 1/2 = 3/2# --> #sin^2 (300) = 3/4#
#sin (300) = +- sqrt3/2#. Since sin (300) is negative then,
#sin (300) = -sqrt3/2#
#tan 300 = sin 300/(cos 300) = (-sqrt3/2)(-2/1) = sqrt3#

Check by calculator.
#tan 300 = 1.73 = sqrt3#.OK

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