AP® Precalculus · Unit 4 · Topic 4.4
Parametrizing a Circle
A circle centered at (h,k) with radius r is traced by x(t) = h + r cos(t), y(t) = k + r sin(t), because cos(t) and sin(t) always satisfy cos2(t) + sin2(t) = 1, locking every point to distance r from the center.
Cosine and sine are not two random wiggly functions paired up. They come from the same angle t on the unit circle, so x = cos(t) and y = sin(t) always obey cos2(t) + sin2(t) = 1. That identity is just the Pythagorean theorem in disguise, and it forces the point (x,y) to sit exactly 1 unit from the origin no matter what t is. Scale both by r and you get a circle of radius r. Shift by adding h to x and k to y, and the center moves to (h,k) without changing the shape.
Once you have x(t) = h + r cos(t) and y(t) = k + r sin(t), every transformation has a visible job. The r stretches the circle's size. The h and k slide the center. The sign in front of t, or a swap between sine and cosine, changes direction and starting point: at t=0, cos(0)=1 and sin(0)=0, so the particle starts at the rightmost point (h+r, k) by default. Flipping the sign of sin(t) reverses direction to clockwise. Swapping which function multiplies x and which multiplies y moves the starting point to the top, bottom, or wherever you need.
This topic is the bridge between the unit's earlier parametric curves and the vector work coming next. A circle traced by (x(t), y(t)) is really a position vector sweeping around, and its rate of change will become a velocity vector later in the unit. Getting comfortable now with how r, h, k, and the cos/sin roles control the picture makes that jump to vectors much smoother.
Key terms
- Parametrization
- A way of writing a curve's x and y coordinates as separate functions of a third variable, usually t.
- Pythagorean identity
- The identity cos2(t) + sin2(t) = 1, true for every value of t.
- Parameter
- The input variable, often t, that generates each point on the curve as it varies.
- Orientation
- The direction a parametrized curve is traced as t increases, such as clockwise or counterclockwise.
Watch it explained
- How it worksHow do transformations of cosine and sine move, resize, and re-time a circular path?
- Worked exampleFind parametric equations for a circle of radius 5 centered at (2, -3) traveled clockwise.
- Going furtherWhat changes in the equations when you want the particle to start at the top of the circle?
Check yourself
End of topic.