Numbers & Functions

6 Trigonometric Functions

CHAPTER 6 Trigonometric Functions Every machine we’ve built so far has had one thing in common: feed it a bigger and bigger input, and the output runs off somewhere and never comes back. Polynomials shoot to infinity. Exponentials explode or decay to nothing. They’re all machines of escape.

This chapter is about machines that do the opposite. You feed them a bigger and bigger input, and the output just. . . goes around. It comes back to where it started, and does it again, forever. These are the machines of repetition — of anything that cycles, oscillates, waves, or spins. The tides. A heartbeat. A sound wave. Alternating current. Light itself.

They’re called the trigonometric functions, and almost everyone first meets them in a confusing way — as ratios of triangle sides, memorized from a chart. We’re going to meet them the right way: as a point walking around a circle.

6.1 The unit circle: one walker, two shadows

Draw a circle of radius 1, centered at the origin. This is the unit circle. Place a walker on the rim at the rightmost point and start them walking counterclockwise. Here’s the whole idea of trigonometry, and it’s just watching the walker’s two shadows: • The walker’s height above the across-axis is the sine of the angle, sin θ. • The walker’s across position is the cosine, cos θ.

That’s it. Sine and cosine are nothing more mysterious than the height and sideways position of a point walking around a circle. Let’s read a few positions straight off the circle: • Start (θ = 0): rightmost point, so cos 0 = 1, sin 0 = 0.

• Quarter turn up: cos = 0, sin = 1.

• Half a turn: cos = −1, sin = 0.

across up 1 P sin θ cos θ θ Figure 6.1.

The unit-circle walker — THE foundational figure of the chapter. At angle θ, the walker’s height is sin θ and its across position is cos θ. Everything else in trigonometry flows from this one picture. • Three-quarters: cos = 0, sin = −1.

Notice the values never leave the range −1 to 1 — the walker is locked onto a circle of radius 1. In Chapter 3 language: both sin and cos have range [−1, 1]. They’re bounded machines, the first we’ve met — utterly unlike the exponentials that ran off to infinity.

6.2 Measuring the angle: why π shows up How do we measure θ? You know degrees — a full turn is 360◦, an ancient, arbitrary choice. Mathematicians use a more natural measure called radians: measure the angle by the distance the walker has actually travelled along the rim. Since the circle has radius 1, its full circumference is 2π. So a full trip is 2π radians, half a trip is π, a quarter turn is π 2 .

This is why π — the circle number from Chapter 2 — suddenly turns up inside angle measurement. It’s the natural unit of “going around.” Trigonometric Functions θ = 0 (1, 0) θ = π 2 (0, 1) θ = π (−1, 0) θ = 3π 2 (0, −1) Figure 6.2. Four key positions, each labelled with its (cos θ, sin θ). Read sine and cosine straight off the circle, no chart: right →(1, 0), top →(0, 1), left →(−1, 0), bottom →(0, −1).

arc length = π 2 ⇒θ = π 2 rad full circumference = 2π Figure 6.3. Radians as arc length. A radian measures the distance walked along the rim; a quarter turn covers arc length π 2 , so the angle is π 2 radians.

π is the natural unit of going around.

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