Solve:4cosθ-3secθ=2tanθ?

1 Answer
Nghi N.
Jun 22, 2018

#x = 5^@74 + k360^@#
#x = 174^@26 + k360^@#
#x = 216^@87 + k360^@#
#x = 323^@13+ k360^@#)

Explanation:

#4cos t - 3/(cos t) = (2sin t)/(cos t)#
Multiply both sides by cos t:
#4cos^2 t - 3 = 2sin t#
#4 - 4sin^2 t - 3 = 2sin t#
#4sin^2 t + 2sin t - 1 = 0#
#D = d^2 = b^2 - 4ac = 4 + 4 = 8# --> #d= +- 2sqrt2#
There are 2 real roots:
#sin x = -b/(2a) +- d/(2a) = - 2/8 +- (2sqrt2)/8 = (-1 +- sqrt2)/4#
#sin x = (-1 + sqrt2)/4 = 0.10#
#sin x = (- 1 - sqrt2)/4 = - 0.60#
a. sin x = 0.10
Calculator and unit circle give 2 solutions for x:
#x = 5^@74#, and #x = 180 - 5.74 = 174^@26#
b. sin x = -0.60
Calculator and unit circle -->
#x = - 36^@87#, and #x = 180 - (-36.87) = 216^@87#
Note x = - 36.87 is co-terminal to #x = 323^@13#
For general answers, add #k360^@#

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