Radians
Why degrees aren't enough
You already know degrees: a full circle is 360°, a right angle is 90°. That works fine for everyday life.
But open any DSP textbook and you'll see angles written as π/2, 2π, or 3π/4. Why? Because radians make the math dramatically simpler, and there's a beautiful reason why.
One Radian = One Radius
Take a circle. Now take its radius and lay it along the edge of the circle (the arc). The angle you've just swept out? That's one radian.
That's all it is: the angle where the arc length equals the radius.
The Important Numbers
Since the circumference of a circle is 2πr, and one radian covers an arc of length r, it takes exactly 2π radians to go all the way around:
| Degrees | Radians | What it means |
|---|---|---|
| 360° | 2π ≈ 6.28 | Full circle |
| 180° | π ≈ 3.14 | Half circle |
| 90° | π/2 ≈ 1.57 | Quarter circle |
| 57.3° | 1 | Arc = radius |
Why DSP Uses Radians
In the previous lesson, we saw that sine and cosine come from a point moving around a circle. In DSP, we describe how fast that point moves as angular frequency, measured in radians per second.
Why not degrees per second? Because with radians:
- The derivative of sin(t) is simply cos(t). With degrees, you'd get an ugly π/180 factor every time.
- Euler's formula ejθ = cos(θ) + j·sin(θ) only works with radians.
- Every frequency formula (like ω = 2πf) is clean and compact.
Converting (When You Must)
Sometimes you still need to convert. The rules are simple:
radians = degrees × π/180
degrees = radians × 180/π
But in DSP, you'll rarely need to. Most formulas are already in radians, and once you get comfortable thinking in terms of π, you won't miss degrees at all.
Frequently Asked Questions
What is a radian?
A radian is the angle at which the arc length along a circle equals the radius. It is the natural unit of angle measurement that emerges from the geometry of the circle itself. One radian is about 57.3 degrees, and a full circle is 2π radians (approximately 6.283).
Why does DSP use radians instead of degrees?
Radians make the math clean. In radians, the derivative of sin(x) is exactly cos(x), with no extra conversion factors. Fourier transforms, transfer functions, and filter equations all assume radians. Using degrees would require multiplying by π/180 everywhere, adding clutter and inviting errors.
Quick Check
Test your understanding of the key concepts from this lesson.
