Signs and Co-signs

Sines and cosines are typically introduced to describe relationships between the sides and angles of right triangles. Sin(x°) = opposite / hypotenuse. Cos(x°) = adjacent / hypotenuse. The hypotenuse is the long side  of the triangle — the one opposite the 90° angle — in case that’s another repressed memory from high school.

Here’s where it gets interesting. As you climb into higher-level math, these functions start showing up everywhere, because it turns out they’re useful for a lot more than triangles. Plot y = sin(x°) and you get a wave — cresting and troughing every 360°. Sin(2x°) oscillates twice as fast, completing a full wave every 180°. That’s frequency. Meanwhile, 2·sin(x°) produces a wave with peaks and valleys twice as tall. That’s amplitude. Two simple knobs: how fast, and how big.

Still with me? Good. Now it gets fun. Or something that looks like fun if you watched Raiders of the Lost Ark on Friday nights in high school while doing your math homework.

Sine and cosine are actually the same function — just shifted 90° apart. Where one crests, the other troughs. They are perpetually, perfectly out of phase.

And when you get deep enough into the math, something kind of insane becomes true: any data set — no matter how chaotic, complex, or seemingly random — can be represented as a sum of sine and cosine waves. You just have to find the right combination of frequencies and amplitudes. It’s called a Fourier transform, for your Tuesday night bar trivia. You don’t need to know how it works. You just need to know that it does — and that it’s kind of miraculous. Given the right mix of waves, you can mathematically reconstruct anything.

Okay. That’s all the nerd speak I’ll inflict on you in one sitting. Now for the reason we went digging in that dusty warehouse.

Life has its own wave patterns. Work, moods, energy, relationships — all of it ebbs and flows in cycles. A crest at work might mean travel, deadlines, late nights, back-to-back meetings. The trough that follows is where you finally exhale — slow mornings, early shutdowns, breathing room. Some of you are already rolling your eyes, convinced your job is a permanent crest with no trough in sight. I’d just say: that implies a very long frequency. The wave is still there. It’ll bottom out eventually — when you quit, when you retire, when something forces a change. Time is infinite. None of us last that long.

I’ve been noticing these patterns more acutely at home lately.

My partner Christine has a lot on her plate. She’s ambitious by nature and commits accordingly — full-time job with regular travel, graduate coursework on top of it, and a genuine talent for stacking her own plate right up to the edge. When everything converges, she’s working weekends and nights. Stress rises. Conversation gets scarce. She knows this about herself, and she leans into it deliberately — because she also knows the trough is coming, and she intends to enjoy it.

Her peaks tend to land in April and November, aligned with the ends of university semesters. Mid-summer and mid-winter are her valleys: more relaxed, more playful, full of entrepreneurial ideas and travel plans. To my thoroughly nerdy eye, it’s a clean sine wave on a six-month cycle.

I’m something like her cosine, at least on these somewhat longer timescales. I split my time between science work and outdoor guiding. Guiding is seasonal — it peaks mid-summer and mid-winter. The shoulder seasons, April and October, go quiet. That’s when I load up the science work. The result is that I tend to be deep in it when she’s surfacing, and surfacing when she’s deep in it. We’re not out of sync. We’re just out of phase.

This shows up on shorter cycles too. We both travel for work, on schedules that seem almost algorithmically designed to keep us moving in opposite directions. She heads to distant corners of Montana every few weeks. My work pulls me out of the country for months, out of state for a week, into the backcountry for days at a time — sometimes on short notice. Lately it feels like we’re always ships passing: I come home as she leaves, we overlap for a day or two, then I head out as she returns. Rinse, repeat.

Layered on top of all that are the higher-frequency waves everyone shares — cooking dinner, walking the dog, paying bills, staying in shape, seeing friends. The small, fast oscillations that fill in the space between the bigger arcs.

Christine and I share a lot — a love of the outdoors, a commitment to community, an entrepreneurial itch we haven’t quite scratched yet. Starting ventures together or supporting each other’s endeavors feels genuinely exciting and daunting in equal measure. There’s always something else competing for the bandwidth. Sometimes it can feel like adding more waves will just make the whole system harder to understand.

But then I think about Fourier transforms.

Any pattern, however complex or chaotic, can be decomposed into waves — and reconstructed from them. You don’t need everything to be in phase. You don’t need the cycles to line up neatly. You just need to find the right combination of frequencies and amplitudes, and let them add up to something intentional.

Out of phase isn’t the same as out of sync.

We’re two waves moving through the same life, cresting and troughing on our own schedules, occasionally missing each other by a half-step. But we’re pointed the same direction. And the math — the actual math — says that’s enough to build anything, if you’re willing to keep solving for the right combination.

I’m still working on the equations. But I think it’s adding up.