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# 18.5: Argument and polar coordinates

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As before, we assume that $$O$$ and $$E$$ are the points with complex coordinates $$0$$ and $$1$$ respectively.

Let $$Z$$ be a point distinct form $$O$$. Set $$\rho=OZ$$ and $$\theta=\measuredangle EOZ$$. The pair $$(\rho,\theta)$$ is called the polar coordinates of $$Z$$.

If $$z$$ is the complex coordinate of $$Z$$, then $$\rho=|z|$$. The value $$\theta$$ is called the argument of $$z$$ (briefly, $$\theta=\arg z$$). In this case,

$$z=\rho\cdot e^{i\cdot\theta}=\rho\cdot(\cos\theta+i\cdot\sin\theta).$$

Note that

$$\arg (z\cdot w) \equiv \arg z+\arg w$$

and

$$\arg \tfrac z w \equiv \arg z-\arg w$$

if $$z \ne 0$$ and $$w \ne 0$$. In particular, if $$Z$$, $$V$$, $$W$$ are points with complex coordinates $$z$$, $$v$$, and $$w$$ respectively, then

\begin{aligned} \measuredangle VZW &=\arg\left(\frac{w-z}{v-z}\right)\equiv \\ &\equiv \arg(w-z)-\arg(v-z) \end{aligned}

if $$\measuredangle VZW$$ is defined.

Exercise $$\PageIndex{1}$$

Use the formula 18.5.1 to show that

$$\measuredangle ZVW+\measuredangle VWZ+\measuredangle WZV\equiv \pi$$

for any $$\triangle ZVW$$ in the Euclidean plane.

Hint

Let $$z, v$$, and $$w$$ denote the complex coordinates of $$Z, V$$, and $$W$$ respectively. Then

$$\begin{array} {rcl} {\measuredangle ZVW + \measuredangle VWZ + \measuredangle WZV} & \equiv & {\arg \dfrac{w - v}{z- v} + \arg \dfrac{z-w}{v-w} + \arg \dfrac{v-z}{w-z} \equiv} \\ {} & \equiv & {\arg \dfrac{(w - v) \cdot (z - w) \cdot (v -z)}{(z - v) \cdot (v - w) \cdot (w - z)} \equiv} \\ {} & \equiv & {\arg (-1) \equiv \pi} \end{array}$$

Exercise $$\PageIndex{2}$$

Suppose that points $$O$$, $$E$$, $$V$$, $$W$$, and $$Z$$ have complex coordinates $$0$$, $$1$$, $$v$$, $$w$$, and $$z=v\cdot w$$ respectively. Show that

$$\triangle OEV\sim \triangle OWZ.$$

Hint

Note and use that $$\measuredangle EOV = \measuredangle WOZ = \arg v$$ and $$\dfrac{OW}{OZ} = \dfrac{OZ}{OW} = |v|$$.

The following theorem is a reformulation of Corollary 9.3.2 which uses complex coordinates.

Theorem $$\PageIndex{1}$$

Let $$\square UVWZ$$ be a quadrangle and $$u$$, $$v$$, $$w$$, and $$z$$ be the complex coordinates of its vertices. Then $$\square UVWZ$$ is inscribed if and only if the number

$$\dfrac{(v-u)\cdot(z-w)}{(v-w)\cdot(z-u)}$$

is real.

The value $$\dfrac{(v-u)\cdot(w-z)}{(v-w)\cdot(z-u)}$$ is called the complex cross-ratio of $$u$$, $$w$$, $$v$$, and $$z$$; it will be denoted by $$(u,w;v,z)$$.

Exercise $$\PageIndex{1}$$

Observe that the complex number $$z\ne 0$$ is real if and only if $$\arg z=0$$ or $$\pi$$; in other words, $$2\cdot\arg z\equiv 0$$.

Use this observation to show that Theorem $$\PageIndex{1}$$ is indeed a reformulation of Corollary 9.3.2.

Hint

Note that

$$\arg \dfrac{(v-u) \cdot (z-w)}{(v -w) \cdot (z -u)} \equiv \arg \dfrac{v - u}{z - u} + \arg \dfrac{z- w}{v -w} = \measuredangle ZUV + \measuredangle VWZ.$$

The statement follows since the value $$\dfrac{(v - u) \cdot (x - w)}{(v - w) \cdot (z - u)}$$ is real if and only if

$$2 \cdot \arg \dfrac{(v - u) \cdot (z - w)}{(v - w) \cdot (z - u)} \equiv 0.$$