When do you use sine or cosine for vector components?

What it looks like

Both rows below have a 50.0 N force and a 30.0° angle. The only change is where the angle is measured from.

The right answers trade places between the rows. In the first row the angle opens from the horizontal, so the horizontal component is next to it and gets the cosine. In the second row the angle opens from the vertical, so the vertical component gets the cosine and Fx gets the sine.

OpenStax University Physics writes Ax = A cos θ and Ay = A sin θ, with θ measured counterclockwise from the positive x-axis. Those two formulas hold only for an angle measured that way.

In alai, an iPad app that reads your handwriting as you work, Ally, the tutor, would point at the sin or cos in a swapped line and ask about the angle.

Why it happens

Components are usually taught first with angles measured from the positive x-axis, where x really does get the cosine. The rule then sticks as a fact about x and y. It's really a fact about which side of the triangle touches the angle.

Problems give angles in other ways too: from the vertical, east of north, below the horizontal, or up a ramp. Each one changes which component is next to the angle.

Ramps make the swap especially easy. On a ramp tilted at θ, the part of the weight along the ramp is mg sin θ and the part pressing into the ramp is mg cos θ. That feels backward until you try a flat ramp: at θ = 0°, sin 0° = 0, so nothing pulls a block along a level floor, which is right.

How to catch it yourself

Imagine the angle shrinking to zero, and ask:

If that angle were 0°, what should your x-component be?

Then put 0° into the formula you wrote. If the formula and the picture disagree, the sine and cosine are in the wrong places.

Try these

1. A ball is kicked at 20.0 m/s, 60.0° above the horizontal. Find vx and vy.
2. A 40.0 N force points 20.0° east of north. Find its east and north components.
3. A 5.00 kg block sits on a ramp tilted 30.0° above the horizontal. Using g = 9.80 m/s², find the parts of its weight along the ramp and into the ramp.

Answers

1. vx = 20.0 cos 60.0° = 10.0 m/s and vy = 20.0 sin 60.0° = 17.3 m/s
2. The angle is measured from north, so north gets the cosine: north = 40.0 cos 20.0° = 37.6 N and east = 40.0 sin 20.0° = 13.7 N.
3. The weight is mg = 49.0 N. Along the ramp it's 49.0 sin 30.0° = 24.5 N, and into the ramp it's 49.0 cos 30.0° = 42.4 N.

How alai helps

alai is a notes app for iPad and the web. In Homework help, its tutor, Ally, follows every handwritten step, and if a sine and cosine are swapped, Ally points at that component and asks a question about the angle, so the fix is still yours. You can also hold to talk and ask Ally to draw a vector and its components on the board beside your work.