Finding Angles

From Rectangular to Polar

Every complex number carries two descriptions. The rectangular form $$z = x + iy$$ names a point; the polar form $$r e^{i\theta}$$ names a distance and a rotation.

Finding angles is the step that connects them.

On the Argand diagram, $$x$$ is the horizontal coordinate and $$y$$ the vertical one. The angle $$\theta$$ measures rotation from the positive real axis.

The Defining Relation

The argument of $$z = x + iy$$ satisfies

$$\tan(\theta) = \frac{y}{x}$$

This comes straight from the right triangle with legs $$x$$ and $$y$$ on the Argand diagram.

Notice that the same right triangle legs x and y that define the tangent ratio also fix the rotation theta from the positive real axis.

But the relation defines $$\theta$$ only up to the periodicity of tangent. That is where the trouble starts.

Why Tangent Alone Fails

The tangent function has period $$\pi$$. So $$\tan(\theta)$$ cannot distinguish an angle from that angle plus $$\pi$$.

The ratio $$y/x$$ is identical for $$z = 1 + i$$ and $$z = -1 - i$$, yet these points sit in opposite quadrants.

A naive arctangent collapses both onto the same answer. You must read the signs of $$x$$ and $$y$$ separately.

The atan2 Function

The four-quadrant inverse tangent formalizes the sign bookkeeping. Written $$\mathrm{atan2}(y, x)$$, it takes both coordinates rather than their ratio.

It returns a unique angle in the interval $$(-\pi, \pi]$$.

That interval is the principal value of the argument. One point, one answer.

Why the Interval Matters

The choice of $$(-\pi, \pi]$$ is a convention, and it has consequences.

Crossing the negative real axis makes the returned angle jump from near $$\pi$$ to near $$-\pi$$. Nothing about the point changed; only the labelling did.

That discontinuity is the branch cut. The behaviour of the angle determines whether a function stays continuous across the complex domain.

Where This Skill Is Used

Accurate arguments are not an exercise in bookkeeping. They are prerequisites.

Multi-valued functions, contour integration, and branch cuts all depend on placing $$\theta$$ correctly.

Get the quadrant wrong and every downstream polar computation inherits the error.

Key Takeaways

The argument converts coordinates into rotation.

Tangent alone loses quadrant information because its period is $$\pi$$; atan2 recovers it by reading both signs and returning a unique angle in $$(-\pi, \pi]$$.

Master this, and polar form, branch cuts, and contour work all rest on solid ground.

The argument of a complex number measures rotation from the positive real axis.

• True
• False

Answer: True

The argument is the angle θ measured counterclockwise from the positive real axis on the Argand diagram.

For $$z=x+iy$$, use _____(y, x) rather than a naive _____ of y/x, because the two-coordinate function preserves quadrant information.

Answer: For $$z=x+iy$$, use atan2(y, x) rather than a naive arctangent of y/x, because the two-coordinate function preserves quadrant information.

atan2 receives y and x separately, while a naive arctangent receives only their ratio and can confuse opposite quadrants.

What is the principal argument of $$z=-1+i$$?

• $$-3\pi/4$$
• $$-\pi/4$$
• $$\pi/4$$
• $$3\pi/4$$

Answer: $$3\pi/4$$

The point has x = -1 and y = 1, so it lies in Quadrant II. Therefore, $$\mathrm{atan2}(1,-1)=3\pi/4$$, which lies in the principal interval $$(-\pi,\pi]$$.