Circles and ellipses in parametric form

Warm-up

Three quick questions from earlier lessons.

Discover

A point moves on $x = \cos t$ , $y = \sin t$ , starting at $(1, 0)$ when $t = 0$ . Where is it at $t = \dfrac{\pi}{2}$ ?

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

Key idea

Cosine and sine make points go around.

x = a\cos t

,

y = b\sin t

is an ellipse with horizontal half-axis

a

and vertical half-axis

b

(a circle if

a = b

), traced counterclockwise from

(a, 0)

. With

\cos(kt)

and

\sin(kt)

, one turn takes

t = \dfrac{2\pi}{k}

.

Worked example

Describe $x = 4\cos t$ , $y = 2\sin t$ .

x

from

-4

to

4

: horizontal half-axis

4

cos t goes from −1 to 1.

y

from

-2

to

2

: vertical half-axis

2

sin t also.

\dfrac{x^2}{16} + \dfrac{y^2}{4} = 1

, an ellipse

cos² + sin² = 1.

Your turn, guided

Your turn: $x = 5\cos(3t)$ , $y = 5\sin(3t)$ .

A circle of radius

5

Same coefficient on both.

One turn takes

t = \dfrac{2\pi}{3}

3t reaches 2π at t = 2π/3.

Practice ★

The ellipse $x = 3\cos t$ , $y = 4\sin t$ reaches $x$ -values from $-c$ to $c$ . What is $c$ ?
The ellipse $x = 7\cos t$ , $y = 5\sin t$ reaches $x$ -values from $-c$ to $c$ . What is $c$ ?
The ellipse $x = 2\cos t$ , $y = 4\sin t$ reaches $x$ -values from $-c$ to $c$ . What is $c$ ?

Watch out!

Practice ★★

A point moves on $x = 7\sin t$ , $y = 7\cos t$ . At $t = 0$ it is at $(0, 7)$ . Which way does it go as $t$ increases?
1. Clockwise
2. Counterclockwise
3. Along a straight line

A point moves on $x = 8\cos(4t)$ , $y = 8\sin(4t)$ . How long does one full turn take?
A point moves on $x = 7\cos(3t)$ , $y = 7\sin(3t)$ . How long does one full turn take?
A point moves on $x = 7\cos(2t)$ , $y = 7\sin(2t)$ . How long does one full turn take?

Exit ticket

Three questions to check you have got it.

A point moves on $x = 8\cos(4t)$ , $y = 8\sin(4t)$ . How long does one full turn take?
A point moves on $x = 7\cos(3t)$ , $y = 7\sin(3t)$ . How long does one full turn take?
A point moves on $x = 7\cos(2t)$ , $y = 7\sin(2t)$ . How long does one full turn take?

Summary card

Circles and ellipses

$x = a\cos t$ , $y = b\sin t$ : half-axes $a$ (horizontal) and $b$ (vertical).
Counterclockwise from $(a, 0)$ ; swapping $\sin$ and $\cos$ reverses it.
$\cos(kt)$ , $\sin(kt)$ : one turn in $\dfrac{2\pi}{k}$ .

Watch out: the number with $\cos$ in $x$ is the horizontal half-axis.

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