Parametric Circles and Lines — AP® Precalculus

Part 1 · Hook

Why this matters

A seat on a Ferris wheel, the tip of a clock hand and a satellite in a round orbit all trace circles, and a drone flying from one landing pad to another traces a straight segment. Parametric equations describe both, and they also say where the trip starts, which way it goes and how long it takes.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What are the coordinates of the point on the unit circle at angle $\frac{\pi}{2}$ ?

$(0, 1)$
$(1, 0)$
$(-1, 0)$
$(0, -1)$

Show the answer

$(\cos\frac{\pi}{2}, \sin\frac{\pi}{2})$ $= (0, 1)$ , a quarter turn counterclockwise from $(1, 0)$ .

Correct: $(0, 1)$ :
$(1, 0)$ :
$(-1, 0)$ :
$(0, -1)$ :

2. What is the slope of the line through $(1, 4)$ and $(5, 1)$ ?

$-\frac{3}{4}$
$-\frac{4}{3}$
$\frac{3}{4}$
$-3$

Show the answer

$\frac{1 - 4}{5 - 1} = \frac{-3}{4}$ .

Correct: $-\frac{3}{4}$ :
$-\frac{4}{3}$ :
$\frac{3}{4}$ :
$-3$ :

3. What is the center of the circle $(x - 2)^2 + (y + 1)^2 = 9$ ?

$(2, -1)$
$(-2, 1)$
$(2, 1)$
$(3, 3)$

Show the answer

$(x - h)^2 + (y - k)^2 = r^2$ has center $(h, k)$ ; $y + 1 = y - (-1)$ , so $k = -1$ .

Correct: $(2, -1)$ :
$(-2, 1)$ :
$(2, 1)$ :
$(3, 3)$ :

Part 4 · See it

See it first

Left: the circle x(t) = 2 cos t, y(t) = 2 sin t. The radius to the point at t = pi/3 makes angle t with the positive x-axis; the trip starts at (2, 0) when t = 0 and arrows show counterclockwise travel. Right: the segment x(t) = 1 + 4t, y(t) = 4 minus 3t from (1, 4) at t = 0 to (5, 1) at t = 1, with the midpoint (3, 2.5) at t = 0.5. — Left: $(2\cos t, 2\sin t)$ starts at $(2, 0)$ and goes counterclockwise; $t$ is the angle. Right: $(1 + 4t, 4 - 3t)$ for $0 \le t \le 1$ goes from $(1, 4)$ to $(5, 1)$ . LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

$x(t) = h + r\cos t$ and $y(t) = k + r\sin t$ every point is $r$ units from $(h, k)$ , so the curve is a circle with center $(h, k)$ and radius $r$
$t$ runs from $0$ to $2\pi$ the point starts at $(h + r, k)$ and goes once around counterclockwise
Replace $\sin t$ with $-\sin t$ the same circle from the same start, but clockwise
$x$ and $y$ are both linear functions of $t$ the points lie on a line whose slope is the rate of $y$ over the rate of $x$

Part 6 · Key ideas

Key ideas

A parametric circle: $x(t) = h + r\cos t$ , $y(t) = k + r\sin t$ , $0 \le t \le 2\pi$ , traces the circle with center $(h, k)$ and radius $r$ once, counterclockwise from $(h + r, k)$ .
Change the direction with $k - r\sin t$ ; change the speed with $\cos(bt)$ and $\sin(bt)$ ; trace an arc with a shorter interval of $t$ .
A segment from $(x_1, y_1)$ to $(x_2, y_2)$ : $x(t)$ $= x_1 + (x_2 - x_1)t$ , $y(t)$ $= y_1 + (y_2 - y_1)t$ , $0 \le t \le 1$ .
If $x(t) = x_0 + at$ and $y(t) = y_0 + bt$ with $a \ne 0$ , the line has slope $\frac{b}{a}$ .

Part 7 · Misconception

A common mistake

The wrong idea: $(3 + 2\cos t, 1 + 2\sin t)$ is a circle of radius $3$ centered at $(2, 1)$ .

What actually happens: The number added is the center's coordinate and the number multiplying $\cos t$ and $\sin t$ is the radius: center $(3, 1)$ , radius $2$ .

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

A circle with a starting point

A point travels once around the circle shown for $0 \le t \le 2\pi$ , starting at the marked point and moving counterclockwise.

A circle with center (2, 1) and radius 3. The trip starts at (5, 1), marked t = 0, and arrows show counterclockwise travel. The center is marked. — LevlPrep original diagram.

1. Which parametric function traces the circle as shown?

$(2 + 3\cos t, 1 + 3\sin t)$
$(2 + 3\cos t, 1 - 3\sin t)$
$(2 + 3\sin t, 1 + 3\cos t)$
$(2 + 9\cos t, 1 + 9\sin t)$

Show the answer

The center is $(2, 1)$ and the radius is $3$ . $(2 + 3\cos t, 1 + 3\sin t)$ starts at $(5, 1)$ when $t = 0$ and moves counterclockwise.

Correct: $(2 + 3\cos t, 1 + 3\sin t)$ : Center $(2, 1)$ , radius $3$ , start $(5, 1)$ , counterclockwise.
$(2 + 3\cos t, 1 - 3\sin t)$ : With $1 - 3\sin t$ the point moves down first from $(5, 1)$ : clockwise.
$(2 + 3\sin t, 1 + 3\cos t)$ : This starts at $(2, 4)$ , the top, and moves clockwise.
$(2 + 9\cos t, 1 + 9\sin t)$ : The radius multiplies $\cos t$ and $\sin t$ ; $9$ is the radius squared.

2. Where is the point when $t = \frac{\pi}{2}$ ?

$(2, 4)$
$(-1, 1)$
$(2, -2)$
$(5, 4)$

Show the answer

$x$ $= 2 + 3\cos\frac{\pi}{2}$ $= 2 + 0$ $= 2$ and $y$ $= 1 + 3\sin\frac{\pi}{2}$ $= 1 + 3$ $= 4$ : the top of the circle, a quarter turn counterclockwise from the start.

Correct: $(2, 4)$ : A quarter turn counterclockwise from $(5, 1)$ is the top, $(2, 4)$ .
$(-1, 1)$ : $(-1, 1)$ is at $t = \pi$ , half a turn.
$(2, -2)$ : $(2, -2)$ is at $t = \frac{3\pi}{2}$ , the bottom.
$(5, 4)$ : $x = 2 + 3\cos\frac{\pi}{2} = 2$ , not $5$ .

3. For which values of $t$ is the point on the upper half of the circle, $y \ge 1$ ?

$0 \le t \le \pi$
$\frac{\pi}{2} \le t \le \frac{3\pi}{2}$
$\pi \le t \le 2\pi$
$0 \le t \le \frac{\pi}{2}$

Show the answer

$y = 1 + 3\sin t \ge 1$ exactly when $\sin t \ge 0$ , which on $[0, 2\pi]$ is $0 \le t \le \pi$ .

Correct: $0 \le t \le \pi$ : $\sin t \ge 0$ on $[0, \pi]$ .
$\frac{\pi}{2} \le t \le \frac{3\pi}{2}$ : On this interval $\cos t \le 0$ : the left half.
$\pi \le t \le 2\pi$ : Here $\sin t \le 0$ : the lower half.
$0 \le t \le \frac{\pi}{2}$ : This is only the upper right quarter.

4. Which change makes the point travel clockwise around the same circle, starting at the same point?

Replace $1 + 3\sin t$ with $1 - 3\sin t$
Replace $2 + 3\cos t$ with $2 - 3\cos t$
Let $t$ run from $0$ to $4\pi$ instead
Replace the $3$ with $-3$ in both coordinates

Show the answer

With $y = 1 - 3\sin t$ the start is still $(5, 1)$ , but just after $t = 0$ the point moves down: clockwise.

Correct: Replace $1 + 3\sin t$ with $1 - 3\sin t$ : $x$ is unchanged, so the start stays $(5, 1)$ , and the point moves down first.
Replace $2 + 3\cos t$ with $2 - 3\cos t$ : This starts at $(-1, 1)$ , a different point.
Let $t$ run from $0$ to $4\pi$ instead: This goes around twice, still counterclockwise.
Replace the $3$ with $-3$ in both coordinates: This starts at $(-1, 1)$ and still goes counterclockwise.

5. How does $(\cos t, \sin t)$ for $0 \le t \le 2\pi$ trace the unit circle?

It starts at $(1, 0)$ and goes once around counterclockwise
It starts at $(0, 1)$ and goes once around clockwise
It starts at $(1, 0)$ and goes once around clockwise
It starts at $(0, 1)$ and goes once around counterclockwise

Show the answer

At $t = 0$ the point is $(\cos 0, \sin 0) = (1, 0)$ ; at $t = \frac{\pi}{2}$ it is $(0, 1)$ . Right to top is counterclockwise.

Correct: It starts at $(1, 0)$ and goes once around counterclockwise: $(1, 0)$ then $(0, 1)$ : counterclockwise.
It starts at $(0, 1)$ and goes once around clockwise: $(0, 1)$ is the start of $(\sin t, \cos t)$ .
It starts at $(1, 0)$ and goes once around clockwise: From $(1, 0)$ the point rises toward $(0, 1)$ , which is counterclockwise.
It starts at $(0, 1)$ and goes once around counterclockwise: At $t = 0$ the point is $(1, 0)$ , not $(0, 1)$ .

6. Describe the trip traced by $(4\sin t, 4\cos t)$ for $0 \le t \le 2\pi$ .

It starts at $(0, 4)$ and goes once around clockwise
It starts at $(4, 0)$ and goes once around counterclockwise
It starts at $(0, 4)$ and goes once around counterclockwise
It starts at $(4, 0)$ and goes once around clockwise

Show the answer

At $t = 0$ : $(0, 4)$ , the top. At $t = \frac{\pi}{2}$ : $(4, 0)$ , the right side. Top to right is clockwise.

Correct: It starts at $(0, 4)$ and goes once around clockwise: Top, then right: clockwise.
It starts at $(4, 0)$ and goes once around counterclockwise: That describes $(4\cos t, 4\sin t)$ ; here sine comes first.
It starts at $(0, 4)$ and goes once around counterclockwise: From the top the point moves to the right side, which is clockwise.
It starts at $(4, 0)$ and goes once around clockwise: At $t = 0$ , $x = 4\sin 0 = 0$ , so the start is $(0, 4)$ .

Part 9 · Summary

Summary

(h + r\cos t, k + r\sin t)

for

0 \le t \le 2\pi

traces the circle with center

(h, k)

and radius

r

once counterclockwise, starting at

(h + r, k)

. A segment from

P

to

Q

is

P + (Q - P)t

for

0 \le t \le 1

, coordinate by coordinate. Both describe not only a set of points but a trip along it.

Part 10 · Up next

What comes next

Part 11 · Connections