How to Sail Upwind (with Trigonometry)

Up here in Boston, you'll see a lot of sailboats out on the Charles river in the fall. (We also just hosted the Head of the Charles, a major annual rowing event.) In sailing, there are all sorts of terminologies and rules, with words like tacking, jibing, and beating. Sailboats can travel upwind, which is pretty amazing when you think about it. But they can't travel completely against the wind -- they "beat" the wind by traveling at a slight angle to the wind. What's going on here?

Let's start off with what a sailboat looks like:

The boat is headed in one direction, its sails are facing a different direction, and there's wind blowing in some third direction (although you can't actually see the wind in the picture). Using the angles between these three directions, and some trigonometry, we'll discover how boats can actually sail upwind.

To help us out with the math, let's draw a simplified version of a sailboat, from a top-down perspective (see the picture below). Suppose that the wind (with strength W) is blowing in a particular direction, that the sails are set an angle θ from the wind, and that the boat is facing a direction that's an additional angle φ from the direction of the sails.

Because the sails are set at an angle from the wind, they won't feel the full strength of the wind. Think of it this way: take a piece of paper, hold it so it faces you, and blow on it -- it will, of course, move. But if you blow on the paper's edge instead, you'll have a much harder time moving it. The same thing happens with sails, and only the perpendicular component of the wind will actually push the boat. Let's break the wind down into components that are parallel and perpendicular to the sails:

In the above picture, the red component of the wind is parallel to the sails, and won't push them at all. But the blue component is perpendicular, and will push the sails down and to the right. As you might know from our lesson on trig functions, if the wind is blowing with strength W, then that perpendicular component has strength Wsin(θ). So if the sails are parallel to the wind, they won't get any push bceause sin(0°) = 0, and if the sails are perpendicular to the wind, they'll get the full force because sin(90°) = 1.

Now sailboats can only travel in the direction they're pointing (that's what rudders and keels are for). So if a sailboat is getting a push, only the component of the push in the direction of the boat will actually move it. A strong push in the perpendicular direction, on the other hand, wouldn't move the boat, but could topple or capsize the boat. We said the push on the sails was Wsin(θ), but now we again break down this force into components to find the component that pushes the boat.

Because of the rudder, the boat can only move forward (or backward), but not sideways. So the red component of the push from the sails won't move the boat. The blue component, however, will move the boat. And again, using trig functions, the blue component has a length of Wsin(θ)sin(φ).

So if the sails are an angle θ from the wind's direction, and the boat is an angle φ from the sails, then the boat can actually travel upwind, with a force that's proportional to sin(θ)sin(φ). The angle between "upwind" and the boat is θ+φ, so if this sum is less than 90°, then the boat is "beating" the wind. But as these angles get smaller, sin(θ)sin(φ) also gets smaller. That means the more you try to sail directly against the wind, the slower you'll go. Typically, the furthest upwind a sailboat can travel is about 35° to 45°.

And one other thing -- we assumed here that the direction of the sails was between the direction of the wind and the direction the boat was facing. Compare these two pictures below:

On the left is our sailboat with the sails between the wind and the boat's direction. As we just discovered, this boat can "beat" the wind. But for the boat on the right, the sails are a greater angle from the wind than the boat is. We could carefully work through the trigonometry again to see what happens, and we'd find that having the sails on the other side of the boat is equivalent to replacing φ with −φ in our previous work. That means the wind is pushing the boat forward with a force that's proportional to sin(θ)sin(−φ), which, by the trig identities for negative angles, is equivalent to −sin(θ)sin(φ). But for typical angles of θ and φ, that's a negative number -- so the boat on the right is in fact being pushed backward by the wind! So if you intend to sail upwind, make sure the sails are always pointing between the direction the wind is coming from and the direction your boat is facing.