#4sin(2x)=tanx#, solve for x (for #0<=x<=360#). I'm getting 7 answers, but only 4 answers are listed, and they are between #0# and #180#. Can somebody solve this?

1 Answer
Nghi N.
Oct 13, 2015

Solve #4sin 2x = tan x#

Ans: #0; pi; 2pi; +- 69^@23#

Explanation:

4sin 2x - tan x = 0.
#4sin 2x - sin x/(cos x) = 0#
4sin 2x.cos x - sin x = 0
8sinx.cos^2 x - sin x = 0
(condition #cos x != 0 -> x != pi/2 and x != (3pi)/2#)
sin x(8cos^2 x - 1) = 0.
a. sin x = 0 --> #x = 0; x = pi; and x = 2pi#
b. #cos^2 x = 1/8# --> #cos x = +- 1/(2sqrt2) = +- 0.35#
a. cos x = 0.35 --> #x = +- 69^@23#
b. cos x = - 0.35 --> #x = +- 110^@49#
Check by calculator.
x = 69.33 --> sin 2x = 0.66 --> 4sin 2x = 2.64
tan x = tan 69.33 = 2.64. OK
x = 110.49 --> sin 2x = sin (220.98) = -0.66 --> 4sin 2x = -2.64 --> tan 110.49 = -2.67. OK

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