Lösung 4.4:5b
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Version vom 15:16, 22. Okt. 2008
Let's first investigate when the equality
| \displaystyle \tan u=\tan v |
is satisfied. Because \displaystyle \tan u can be interpreted as the slope (gradient) of the line which makes an angle u with the positive x-axis, we see that for a fixed value of \displaystyle \tan u, there are two angles v in the unit circle with this slope,
| \displaystyle v=u\qquad\text{and}\qquad v=u+\pi\,\textrm{.} |
The angle v has the same slope after every half turn, so if we add multiples of \displaystyle \pi to u, we will obtain all the angles v which satisfy the equality
| \displaystyle v=u+n\pi\,, |
where n is an arbitrary integer.
If we apply this result to the equation
| \displaystyle \tan x=\tan 4x |
we see that the solutions are given by
| \displaystyle 4x = x+n\pi\qquad\text{(n is an arbitrary integer),} |
and solving for x gives
| \displaystyle x = \tfrac{1}{3}n\pi\qquad\text{(n is an arbitrary integer).} |
