Question
How to convert $\arctan\left(\frac{4}{3}\right)$ into rectangular form?
I know that tan=$x/y$ but in that case this would be useless and I was wondering how to proceed.
How to convert $\arctan\left(\frac{4}{3}\right)$ into rectangular form
1
$\begingroup$
calculus
algebra-precalculus
complex-numbers
polar-coordinates
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5What do you mean by `convert into rectangular form`? FWIW $\alpha= \arctan(4/3)$ is not a "nice" value (though $\sin(\alpha)$ and $\cos(\alpha)$ *are*). – 2017-02-06
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0Do you actually have the polar *equation* $\theta = \arctan(4/3)$? – 2017-02-06
2 Answers
1
The polar equation
$$ \theta=\arctan\left(\frac{4}{3}\right)$$
may be converted into a rectangular equation as follows:
$\tan\theta=\dfrac{4}{3}$
$\dfrac{\sin\theta}{\cos\theta}=\dfrac{4}{3}$
$3\sin\theta=4\cos\theta$
$3r\sin\theta=4r\cos\theta$
$3y=4x$
$y=\dfrac{4}{3}x$
Is this what you had in mind?
0
What is the rectangular form of a number? Anyway, $$ \arctan\frac{4}{3}=\arcsin\frac{4}{5}=\arccos\frac{3}{5} $$ is not a rational multiple of $\pi$, by the following well-known lemma: $$ q\in\mathbb{Q},\;\cos(\pi q)\in\mathbb{Q}\quad\Longrightarrow\quad \cos(\pi q)\in\left\{-1,-\frac{1}{2},0,\frac{1}{2},1\right\}.$$ Numerically, $\arctan\frac{4}{3}\approx 53^\circ 7'48''$.