Sine, Cosine & Calculus: The Spell Behind the Falling Moon, AI and August Heat | Math in Use

The Moon, still falling toward Earth. A 660-ton pendulum guarding a skyscraper. Phone video that doesn't shake, even while you're running. The way AI keeps track of word order. All four come down to sine and cosine. But they need a partner: calculus. At school, we learned them as separate topics. So how can they team up? Only one kind of wave keeps its shape when you differentiate it: sine and cosine. Differentiate them, and they just shift by 90 degrees. Other waves lose their shape. A triangle wave becomes a square wave, a square wave becomes nothing but spikes, and a sawtooth becomes a flat line with spikes. That's why sine and cosine are perfect partners for calculus. The longest day of the year comes in June. But the hottest month is August. And the days lengthen fastest in March, around the spring equinox (in Tokyo, by about two minutes a day). March, June, August: three months that seem unrelated, all tied together by one spell: sine, cosine, derivative, integral. Add a time axis to a point going around a circle, and it traces a helix. The whole is a helix. A piece is a wave. An instant is a line (the derivative). Add the lines up, and it's a wave again (the integral). And differentiating the helix simply turns it 90 degrees. This video follows our episodes on sine and cosine and on derivatives and integrals, but you don't need to have seen them. We focus on how the math is used, not how to solve it. ▼ In this video ・One helix holds it all: the circle, the wave, the slope (derivative), and the running total (integral) ・Differentiate, and the helix turns 90 degrees (sine → cosine → minus sine…). Integrate, and it turns back (sine → minus cosine → minus sine…) ・When sine is zero, cosine is at a peak or a trough. When sine is at a peak or a trough, cosine is zero ・Why exactly 90 degrees? For a point going around a circle, the arrow of its motion always points 90 degrees ahead ・Technique 1, reading ahead (derivative): the peak of change comes first. Ferris wheels, day length (March and June), the tides' "rule of twelfths", seasonal sales ・Technique 2, reading the delay (integral): the peak of the running total comes later. A bathtub whose faucet is set by the Sun, heat in minus heat out, August heat, the afternoon high, the coldest time of year, store inventory, the 600-degree rule for cherry blossoms ・Technique 3, reading the swing (differentiate twice): a spring's position, speed and pull are three waves 90 degrees apart. Resonance and pushing in rhythm, Taipei 101's giant pendulum, radio, a quartz watch's 32,768 vibrations a second ・Applications: the Moon is always falling toward Earth (seen edge-on, it swings like a spring; so why doesn't it hit us? Newton's cannon and a satellite's 7.9 km per second), phone image stabilization, VR headsets, and the "rotation" inside AI chatbots ・From sine, cosine, tangent to sine, cosine, calculus: setting sine and cosine free from the triangle ・Cast the spell on AI: ask it to "model the sales as sine and cosine, differentiate, and tell me when the next peak comes", and the calculation starts running ・Bonus: (sin x)′ = cos x, and e^{iθ}: multiplying by i means turning 90° ▼ Chapters 0:00 The Moon, a skyscraper, a phone, and AI: mixing sine and cosine with calculus? 0:27 The only wave that keeps its shape 0:52 Longest day in June, hottest month in August 1:43 Sine, Cosine & Calculus 2:11 Review: the helix and the driver's seat 2:49 One helix holds it all 3:49 Differentiate, and the helix turns 4:34 Sine and cosine side by side: zero meets a peak or a trough 4:50 Why exactly 90 degrees? 5:22 Differentiate again and again: one full turn (integration turns back) 5:56 Technique 1, reading ahead: Ferris wheels and day length 6:41 Reading ahead: tides and seasonal sales 7:55 Technique 2, reading the delay: the bathtub (the Sun controls the faucet) 8:53 Heat in minus heat out, added up = the integral 9:51 Reading the delay: the daily high and the coldest time of year 10:10 Reading the delay: store inventory 10:35 Reading the delay: the 600-degree rule for cherry blossoms 11:04 Technique 3, reading the swing: differentiate twice 11:22 A spring's three waves 11:47 The rhythm of a swing, and resonance 12:21 Pushing in rhythm 12:32 Reading the swing: Taipei 101, radio, and quartz watches 13:25 Application: the Moon is falling 14:04 Seen edge-on, the Moon swings like a spring 14:21 Newton's cannon and satellites 15:15 Application: phone stabilization and the rotation inside AI 16:24 How to use the spell (summary) 17:04 Setting sine and cosine free from the triangle 17:49 Cast the spell on AI 18:27 Bonus: the formulas, and e^{iθ} ▼ Series "Math in Use" We explain math not by how to solve it, but by how it's used in the real world. Now that AI can do the calculating, knowing where and why math is used matters more than ever. Note: Some footage is AI-generated video (KLING). The narration uses an AI voice (ElevenLabs). #trigonometry #calculus #mathinuse