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If cosx = tanx, determine all possible values of sinx.

in not sure how to do this question. I just wrote sinx = cos^2x but I think there is more to this problem.

any ideas

tnx a million

User Iyuna
by
8.1k points

1 Answer

2 votes
If
\cos x=\tan x, then


\cos x=(\sin x)/(\cos x)\implies\cos^2x=\sin x

for
\cos x\\eq0. This is definitely the case, since
\tan x is undefined if
\cos x=0. Now,
\cos^2x=1-\sin^2x, so we have


1-\sin^2x=\sin x\implies\sin^2x+\sin x-1=0\implies\sin x=\frac{-1\pm\sqrt5}2

or
x\approx-1.618 and
x\approx0.618. One of these solutions is larger than 1 in absolute value, but
|\sin x|\le1, so we omit that solution, leaving us with


\sin x=\frac{\sqrt5-1}2\implies x=\arcsin\left(\frac{\sqrt5-1}2\right)+2n\pi

for integers
n, which follows from the fact that
\sin x has period
2\pi.
User Brady Maf
by
8.6k points
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