How do you use the double-angle identities to find tan(2x) if cos x=8/17 and sin x is less than 0?

1 Answer
Oct 18, 2015

Find tan 2x knowing #cos x = 8/17#

Ans: 1.49

Explanation:

#cos x = 8/17#.
Apply the trig identity: #cos^2 x + sin^2 x = 1#
#sin^2 x = 1 - 64/289 = (289 - 64)/289 = 225/289#
#sin x = +-15/17.# Since sin x < 0, therefor:
#sin x = -15/17#
Apply the two trig identities: #sin 2a = 2sin a.cos a#
and #cos 2a = 2cos^2 a - 1#.
#sin 2x = 2sin x.cos x = 2(-15/17)(8/17) = -240/289#
#cos 2x = 2cos^2 x - 1 = 2(64/289) - 1 = 128/289 - 1 =#
#= (128 - 289)/289 = 161/289#.
#tan 2x = (sin 2x)/(cos 2x) = -240/161 = 1.49#

Check by calculator.
#cos x = 8/17# --> #x = 61^@92# --> 2x = 123^#85 --> #tan 2x = 1.49#. OK