Most trig textbooks hand you a page of 30+ identities and say "memorize these." Don't do that.
In DSP, you'll use a handful of identities over and over, and the rest almost never. This lesson covers just those: the ones that actually appear when you're working with signals, filters, and Fourier analysis. For each one, we'll explain where it shows up so you know why it's worth remembering.
Pythagorean Identity
sin²(θ) + cos²(θ) = 1
Where it shows up: Everywhere. This says the point on the unit circle always stays on the circle, so its x² + y² always equals 1. You'll use this to simplify expressions, compute signal power, and prove other identities.
Negative Angle
sin(−θ) = −sin(θ) | cos(−θ) = cos(θ)
Where it shows up: Sine is odd, cosine is even. This matters whenever you work with the Fourier transform, which splits signals into even and odd parts. It's also why the magnitude spectrum is symmetric.
Sine ↔ Cosine Shift
cos(θ) = sin(θ + π/2) | sin(θ) = cos(θ − π/2)
Where it shows up: You saw this in the
Frequency & Phase lesson. Cosine is just sine shifted by π/2. This means any formula written with sine can be rewritten with cosine (or vice versa) by adjusting the phase.
These tell you what happens when you add or subtract angles inside sine and cosine:
Sine of a Sum
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
Where it shows up: Deriving the formula for a phase-shifted sinusoid. When you write sin(ωt + φ), this identity expands it into a weighted sum of sin(ωt) and cos(ωt), which is exactly how the Fourier series works.
Cosine of a Sum
cos(A + B) = cos(A)cos(B) − sin(A)sin(B)
Where it shows up: Same situations as sine-of-a-sum. Also used in rotation matrices. If you've ever rotated a point in 2D, this identity is doing the work.
Tip: You don't need to memorize the difference formulas separately. Just replace B with −B in the sum formulas and use the negative-angle identities: sin(−B) = −sin(B) and cos(−B) = cos(B).
This is the identity that might surprise you with how useful it is. When you multiply two sinusoids, the result is a sum of sinusoids at the sum and difference frequencies:
Cosine × Cosine
cos(A) · cos(B) = ½[cos(A − B) + cos(A + B)]
Sine × Sine
sin(A) · sin(B) = ½[cos(A − B) − cos(A + B)]
Sine × Cosine
sin(A) · cos(B) = ½[sin(A + B) + sin(A − B)]
This is the mathematical basis of modulation in radio, the mixer in a receiver, and how frequency shifting works. Let's see it in action:
Try it: Set f1 = 2 and f2 = 3. The product (green) oscillates at the sum (5) and difference (1) frequencies. You can see the slow "beating" at the difference frequency as an envelope around the fast oscillation. Now try f1 = f2: the difference frequency becomes 0, and you get a steady offset plus a double-frequency wave.
Key Insight: Multiplying two frequencies creates two new frequencies (sum and difference). This is how AM radio works: multiply your audio signal by a high-frequency carrier, and the audio shifts up to radio frequencies. A receiver multiplies again to shift it back down. It's also why you hear "beats" when two nearby piano notes are played together.
These are special cases of the sum formulas where A = B:
Double Angle
sin(2θ) = 2 sin(θ) cos(θ)
cos(2θ) = cos²(θ) − sin²(θ) = 2cos²(θ) − 1 = 1 − 2sin²(θ)
Where it shows up: The cos(2θ) form is especially useful because it rearranges to give you power reduction formulas: cos²(θ) = ½(1 + cos 2θ) and sin²(θ) = ½(1 − cos 2θ). These appear when computing signal power and in the derivation of the DFT.
Here's a secret: once you learn Euler's formula in the next chapter, you won't need to memorize most of these identities. Euler's formula lets you derive them on the fly using simple algebra:
ejθ = cos(θ) + j·sin(θ)
Every identity in this lesson (sum formulas, product-to-sum, double angle, all of them) falls out of Euler's formula with a few lines of algebra. So don't panic about memorization. Understand the identities for now, and the next chapter will give you the master key.
Key Insight: Trig identities aren't arbitrary rules to memorize. They're consequences of the geometry of the circle. And Euler's formula (coming next) turns them all into straightforward algebra. For now, focus on recognizing these identities when they appear, especially product-to-sum.
Cheat Sheet
Here they all are in one place. Bookmark this page if you want:
| Identity |
Formula |
| Pythagorean |
sin²θ + cos²θ = 1 |
| Negative angle |
sin(−θ) = −sinθ, cos(−θ) = cosθ |
| Sine ↔ cosine |
cosθ = sin(θ + π/2) |
| Sine of sum |
sin(A+B) = sinA cosB + cosA sinB |
| Cosine of sum |
cos(A+B) = cosA cosB − sinA sinB |
| Product → sum |
cosA cosB = ½[cos(A−B) + cos(A+B)] |
| Double angle |
sin2θ = 2sinθcosθ, cos2θ = 2cos²θ − 1 |
| Power reduction |
cos²θ = ½(1 + cos2θ), sin²θ = ½(1 − cos2θ) |
Frequently Asked Questions
Which trig identities are most important for DSP?
Three families matter most: (1) the Pythagorean identity sin²θ + cos²θ = 1, used in power calculations; (2) angle addition formulas sin(A±B) and cos(A±B), used in modulation and mixing; (3) product-to-sum identities like cos(A)cos(B) = ½[cos(A-B) + cos(A+B)], which explain why multiplying signals creates sum and difference frequencies.
What is the product-to-sum identity and why does it matter?
The product-to-sum identity converts the multiplication of two sinusoids into a sum of sinusoids at the sum and difference frequencies. For example, cos(A)cos(B) = ½[cos(A-B) + cos(A+B)]. This is the mathematical basis of AM radio, mixer circuits, and frequency translation in every communication system.
Quick Check
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