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Solve for general solutions

$\tan(x/3) = 1$

When I solve this equation my answer comes to be $x = 3\pi/4 \pm 2n\pi, 15\pi/4 \pm 2n\pi$ where $n$ is an integer

However when I graph the equation $y = \tan(x/3) - 1$ values for $x$ such as $11\pi/4$ do not equal zero.

What would $x$ be equal to then?

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3 Answers 3

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The general solution can be described as $$\frac x3\equiv \frac \pi 4\pmod \pi\iff x\equiv \frac{3\pi}4\pmod{3\pi}$$ since the tangent function has period $\pi$.

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Hint:

General solution of $\tan x$ is

$$\tan { x } =a\\ x=\arctan { a } +k\pi ,k\in \mathbb{Z} $$

where is period is $\pi $ not $2\pi $

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$\tan\left(\dfrac{x}{3}\right)=1$

$\dfrac{x}{3}=\dfrac{\pi}{4}+n\pi,\;\forall n\in\mathbb{Z}$

and this becomes

$x=\dfrac{3}{4}\pi+3n\pi,\;\forall n\in\mathbb{Z}$

This explains why solutions are $\dfrac{3}{4}\pi,\;\dfrac{15 \pi }{4},\dfrac{27 \pi }{4},\ldots$ etc

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