It starts with a road sign
A sign warns of a 10% grade. A treadmill offers 30% incline. Both are ratios from chapter one — rise over run — and neither is an angle. Converting between the two is what trigonometry is for, and the function that does it is the tangent.
The chapter takes the unit circle apart in that light. Radians stop being an awkward second unit once you see that an angle is an arc: walk along the rim of a circle of radius 1 and the distance you cover is the angle. No conversion factor to memorise, because there's nothing being converted.
From there, sine is simply how high you climbed on that walk, and cosine is how far across —
which is also why cosine is the sine of the complement, rather than a separate fact to learn.
Tangent is the odd one out, and the chapter gives it its own job: a ruler for
slopes, the height where your walk crosses the vertical line at
x = 1. Which is exactly why every slope percentage on a road sign is a
tangent value, and why the video can go back and answer the sign it opened with.
Chapters in the video
- 0:00The 10% road sign and the 30% treadmill
- 0:35Degrees, radians, and the unit circle
- 1:28Sine is how high you climbed
- 2:39Cosine is the sine of the complement
- 3:56Tangent is a ruler for slopes
- 5:18Back to the road sign
- 5:50Direction is an arc length
Read it instead
The road-sign question, written out on its own page: slope and angle aren't the same number — why a 15% treadmill incline is about 8.5°.