SOLUTION: find all solutions of the equation tan^2theta=3/2sec theta in the interval [0, 2pi)

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Question 454166: find all solutions of the equation tan^2theta=3/2sec theta in the interval [0, 2pi)
Answer by lwsshak3(11628) About Me  (Show Source):
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find all solutions of the equation tan^2theta=3/2sec theta in the interval [0, 2pi)
..
tan^2x=(3/2)secx
identity: tan^x+1=sec^2x
sec^2x-1=3secx/2
2sec^2x-2=3secx
2sec^2x-3secx-2=0
solve by quadratic formula below:
..
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
..
a=2, b=-3, c=-2
secx=[-(-3)ħsqrt((-3)^2-4*2*-2)]/2*2
=[3ħsqrt(9+16)]/4
=(3ħ√25)/4
=(3ħ5)/4
=8/4=2
or
=-2/4=-1/2
..
secx=2
cosx=1/2
x=π/3 and 5π/3
or
secx=-1/2
cosx=-2 (reject, (-1 < cosx < 1)
ans:
x=π/3 and 5π/3