I am stuck when it comes to finding the end value of a trig function. I have the following question:
$$ \tan^5x - 9\tan{x} = 0 $$
I worked the problem and got:
$$ \tan x = 0\\ \tan^4x-9 = 0\\ x = 0, \pi, \frac {\pi}{3}, \frac {2\pi}{3}, \frac {4\pi}{3}, \frac {5\pi}{3} $$
My book answer is $x = \frac {\pi k}{3}$ how do you get that? I understand that tan uses $ \pi $ and sin, cos use $ 2\pi $ but I'm not sure how they got to that answer.
$\endgroup$ 23 Answers
$\begingroup$$$\tan^4x=9\Longrightarrow \tan^2x= 3\Longrightarrow \tan x=\pm\sqrt 3=\pm\frac{\frac{\sqrt 3}{2}}{\frac{1}{2}}=\pm\frac{\sin\frac{\pi}{3}}{\cos\frac{\pi}{3}}$$
Can you see it now?
$\endgroup$ $\begingroup$$\tan^2x=3\implies \cos2x=\frac{1-\tan^2x}{1+\tan^2x}=\frac{1-3}{1+3}=-\frac12=\cos\frac{2\pi}3$
$2x=2n\pi\pm \frac{2\pi}3=\frac{2\pi}3(3n\pm1)$ where $n$ is any integer.
$x=\frac{\pi}3(3n\pm1)$
Now, we need $0\le \frac{\pi}3(3n+1)<2\pi\implies 0\le3n+1<6\implies n=0,1$
For $n=0, x=\frac{\pi}3(3n+1)= \frac{\pi}3; n=1\implies x=\frac{\pi}3(3n+1)= \frac{4\pi}3$
Similarly, we need $0\le \frac{\pi}3(3n-1)<2\pi\implies 0\le3n-1<6\implies n=1,2$
For $n=1, x=\frac{\pi}3(3n-1)= \frac{2\pi}3; n=2\implies x=\frac{\pi}3(3n-1)= \frac{5\pi}3$
$\endgroup$ $\begingroup$You made an error in your work. e.g. when you solved $\tan x = 0$, you only found the two solutions $0$ and $\pi$. But you missed all of the other solutions like $2 \pi$, $3 \pi$, and $-1872 \pi$. The set of all solutions to $\tan x = 0$ is the set of all values $\pi v$ where $v$ ranges over all integers.
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