Trig derivatives from geometry

What you’ll learn

The power rule handled anything that is a power of t, but sin(x) and cos(x) are not powers of anything, so they need their own derivation. The single capability this lesson builds: derive the trig derivatives, d/dx(sin x) = cos x and d/dx(cos x) = -sin x, from the geometry of a point moving around the unit circle, so you can reconstruct them (and the sign) instead of memorizing them.

You will start from what sine and cosine actually are, the vertical and horizontal coordinates of a point on the unit circle, then use one key fact: in radians the point moves at unit speed, because arc length equals angle. A point circling at unit speed has a velocity perpendicular to its position, of length 1, which is just the position rotated a quarter turn counterclockwise: (cos x, sin x) becomes (-sin x, cos x). Reading off the components gives both derivatives at once, and shows why the minus sign lands on cosine (its coordinate shrinks as the point climbs). You will sanity-check the formulas against the curve shapes, see the small-angle approximation sin(x) ≈ x fall out of the slope at zero, meet the oscillation equation f'' = -f, and learn why radians (not degrees) keep the derivatives free of a stray π/180.

Where this fits

This is lesson 4 of Phase 1 (What a derivative is) and the last of the phase. It applies the same nudge-and-look method as lesson 3’s power rule, just to a point on a circle instead of a growing square, so it reinforces the method while adding the trig derivatives to your toolkit. The small-angle approximation here is the first hint of lesson 13 (Taylor series); the rotation fact (a, b) -> (-b, a) connects back to the Linear Algebra track. Phase 2 then turns to combining functions, with the product rule and the chain rule.

Before you start

Prerequisite (within this track): lesson 3, The power rule from geometry, for the nudge-and-look method (a derivative is how a quantity changes when you nudge its input). It helps to recall from trigonometry that a point on the unit circle at angle x has coordinates (cos x, sin x), and to have seen that rotating a vector (a, b) by 90° counterclockwise gives (-b, a) (covered in the Linear Algebra track). No coding, nothing installed; the practice is pen and paper, with a calculator handy for the small-angle check.

By the end, you’ll be able to

• Derive the derivatives of sine and cosine from a point moving around the unit circle at unit speed
• Explain why the velocity is the position rotated 90° counterclockwise, giving (-sin x, cos x), and why the minus lands on cosine
• Sanity-check the trig derivatives against the shapes of the sine and cosine curves
• Explain why radians are required (degrees introduce a π/180 factor) and state the small-angle approximation sin(x) ≈ x

Time and difficulty

• Read time: about 11 minutes
• Practice time: about 12 minutes (reading the velocity and slopes at a new angle, a small-angle approximation check, and flashcards)
• Difficulty: standard