Parametric equations and polar coordinates

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1. Parametric equations: points and motion

Warm-up

Three quick questions from earlier lessons.

Discover

A drone flies so that after $t$ seconds it is at $x = 2t$ meters east and $y = t + 1$ meters up. Where is it after 3 seconds?

$(6, 4)$
$(4, 6)$

Key idea

Parametric equations give $x$ and $y$ separately, in terms of a third variable $t$ (often time).

The point at time

t

\big(x(t), y(t)\big)

. As

t

increases, the point moves along the curve; the parameter tells when it is where.

Worked example

$x(t) = t^2 - 1$ , $y(t) = 3t$ . Find the point at $t = -2$ .

x(-2) = (-2)^2 - 1 = 3

Replace t by −2 in x, with parentheses.

y(-2) = 3 \cdot (-2) = -6

Same t in y.

The point is

(3, -6)

Both coordinates.

Your turn, guided

Your turn: same curve, $t = 2$ .

x(2) = 3

2² − 1.

y(2) = 6

, so the point is

(3, 6)

3 · 2.

Practice ★

A point moves along $x(t) = t^2 - 4$ , $y(t) = 4t + 2$ . Where is it at $t = 4$ ?
A point moves along $x(t) = t^2 - 7$ , $y(t) = 5t + 2$ . Where is it at $t = 3$ ?
A point moves along $x(t) = t^2 - 5$ , $y(t) = 3t + 2$ . Where is it at $t = 4$ ?

Watch out!

Practice ★★

$x(t) = t + 1$ and $y(t) = t^2$ . Tap the point where the curve is at $t = 1$ .
$x(t) = t + 1$ and $y(t) = t^2$ . Tap the point where the curve is at $t = 0$ .

$x(t) = 4t + 4$ and $y(t) = t - 1$ . At what time $t$ does the point have $x = 8$ ?
$x(t) = 2t + 6$ and $y(t) = t - 1$ . At what time $t$ does the point have $x = 18$ ?
$x(t) = 2t + 1$ and $y(t) = t - 1$ . At what time $t$ does the point have $x = 3$ ?

Exit ticket

Three questions to check you have got it.

$x(t) = 4t + 4$ and $y(t) = t - 1$ . At what time $t$ does the point have $x = 8$ ?
$x(t) = 2t + 6$ and $y(t) = t - 1$ . At what time $t$ does the point have $x = 18$ ?
$x(t) = 2t + 1$ and $y(t) = t - 1$ . At what time $t$ does the point have $x = 3$ ?

Summary card

Parametric equations

The point at time $t$ is $\big(x(t), y(t)\big)$ .
"Where at time $t$ ": evaluate. "When at $x = c$ ": solve $x(t) = c$ .

Watch out: $x$ first, then $y$ .

2. Eliminate the parameter

Warm-up

Three quick questions from earlier lessons.

Discover

$x = t + 2$ and $y = t^2$ . Which equation links $x$ and $y$ directly?

$y = (x - 2)^2$
$y = x^2$

Key idea

Removing $t$ shows the shape of the curve.

To eliminate the parameter: solve one equation for

t

, then substitute into the other. For

x = r\cos t

y = r\sin t

, use

\cos^2 t + \sin^2 t = 1

instead:

x^2 + y^2 = r^2

Worked example

Eliminate $t$ : $x = 3t$ , $y = 6t - 1$ .

t = \dfrac{x}{3}

Solve the simpler one for t.

y = 6 \cdot \dfrac{x}{3} - 1

Substitute into y.

y = 2x - 1

Simplify: a line.

Your turn, guided

Your turn: $x = t - 4$ , $y = t^2$ .

t = x + 4

Add 4.

y = (x + 4)^2

Substitute into y = t². Type y.

Practice ★

$x = t + 6$ and $y = t^2$ . Write $y$ in terms of $x$ .
$x = t + 8$ and $y = t^2$ . Write $y$ in terms of $x$ .
$x = t + 5$ and $y = t^2$ . Write $y$ in terms of $x$ .

Watch out!

Practice ★★

$x = 3t$ and $y = 4t + 9$ . Write $y$ in terms of $x$ .
$x = 2t$ and $y = 2t + 3$ . Write $y$ in terms of $x$ .
$x = 6t$ and $y = 8t + 7$ . Write $y$ in terms of $x$ .

What curve do $x = 5\cos t$ and $y = 5\sin t$ draw, for $0 \le t < 2\pi$ ?
1. A circle of radius 5 centered at the origin
2. A line
3. A circle of radius $25$

Exit ticket

Three questions to check you have got it.

$x = 3t$ and $y = 4t + 9$ . Write $y$ in terms of $x$ .
$x = 2t$ and $y = 2t + 3$ . Write $y$ in terms of $x$ .
$x = 6t$ and $y = 8t + 7$ . Write $y$ in terms of $x$ .

Summary card

Eliminating the parameter

Solve one equation for $t$ .
Substitute into the other.
With $\cos t$ and $\sin t$ : use $\cos^2 t + \sin^2 t = 1$ .

Watch out: solve for $t$ ; do not just replace $t$ by $x$ .

3. 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

4

: horizontal half-axis

4

cos t goes from −1 to 1.

y

from

-2

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.

4. Polar coordinates

Warm-up

Three quick questions from earlier lessons.

Discover

A lighthouse beam points at $90^\circ$ (straight north) and a boat is 3 km away along it. In rectangular coordinates, where is the boat?

$(0, 3)$
$(3, 0)$

Key idea

Polar coordinates $(r, \theta)$ give a point by its distance $r$ from the origin and its angle $\theta$ from the positive $x$ -axis.

x = r\cos\theta

y = r\sin\theta

. Back:

r = \sqrt{x^2 + y^2}

and

\tan\theta = \dfrac{y}{x}

, with

\theta

in the quadrant of the point.

Worked example

Convert $(\sqrt{3}, -1)$ to polar coordinates.

r = \sqrt{3 + 1} = 2

Distance from the origin.

\theta = 2\pi - \dfrac{\pi}{6} = \dfrac{11\pi}{6}

tan θ = −1/√3; the point is right and down: quadrant IV.

\left(2, \dfrac{11\pi}{6}\right)

Both coordinates.

Your turn, guided

Your turn: convert $\left(4, \dfrac{\pi}{3}\right)$ to rectangular.

x = 2

4 cos(π/3) = 4 · 1/2.

y = 2\sqrt{3}

4 sin(π/3) = 4 · √3/2.

Practice ★

Convert the polar point $\left(7, \frac{\pi}{6}\right)$ to rectangular coordinates (exact).
Convert the polar point $\left(8, \frac{5 \pi}{3}\right)$ to rectangular coordinates (exact).
Convert the polar point $\left(7, \frac{2 \pi}{3}\right)$ to rectangular coordinates (exact).

Watch out!

Practice ★★

Convert the point $(-2, 2)$ to polar coordinates $(r, \theta)$ with $r > 0$ and $0 \le \theta < 2\pi$ .
Convert the point $(-5, 5)$ to polar coordinates $(r, \theta)$ with $r > 0$ and $0 \le \theta < 2\pi$ .

Tap the point with polar coordinates $\left(4, \frac{3 \pi}{2}\right)$ .
Tap the point with polar coordinates $\left(3, \pi\right)$ .

Exit ticket

Three questions to check you have got it.

Convert the polar point $\left(7, \frac{\pi}{6}\right)$ to rectangular coordinates (exact).
Convert the polar point $\left(8, \frac{5 \pi}{3}\right)$ to rectangular coordinates (exact).
Convert the polar point $\left(7, \frac{2 \pi}{3}\right)$ to rectangular coordinates (exact).

Summary card

Polar coordinates

$x = r\cos\theta$ , $y = r\sin\theta$ .
$r = \sqrt{x^2 + y^2}$ , $\tan\theta = \dfrac{y}{x}$ .

Watch out: choose $\theta$ in the quadrant of the point.

5. Polar equations and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

Which points satisfy the polar equation $r = 3$ ?

All points at distance 3 from the origin
The points on one line through the origin

Key idea

A polar equation links $r$ and $\theta$ . To recognize it, convert to $x$ and $y$ .

Use

x = r\cos\theta

y = r\sin\theta

and

r^2 = x^2 + y^2

. Classic graphs:

r = a

is a circle around the origin;

\theta = \alpha

is a line through the origin with slope

\tan\alpha

;

r = 2a\sin\theta

is a circle of radius

|a|

centered at

(0, a)

Worked example

Identify $r = 6\cos\theta$ .

r^2 = 6r\cos\theta

Multiply both sides by r.

x^2 + y^2 = 6x

r² = x² + y², r cos θ = x.

(x - 3)^2 + y^2 = 9

: a circle, center

(3, 0)

, radius 3

Complete the square in x.

Your turn, guided

Your turn: $r = 8\sin\theta$ .

x^2 + y^2 = 8y

Multiply by r, convert.

x^2 + (y - 4)^2 = 16

: radius

4

(y − 4)² − 16: type the radius.

Practice ★

What is the graph of the polar equation $r = 5$ ?
1. A circle of radius 5 centered at the origin
2. A line through the origin
3. The vertical line $x = 5$
What is the graph of the polar equation $r = 7$ ?
1. A circle of radius 7 centered at the origin
2. A line through the origin
3. The vertical line $x = 7$

Watch out!

Practice ★★

The polar equation $r = 12\sin\theta$ is a circle. In rectangular coordinates, it is $x^2 + y^2 = 12y$ . What is the $y$ -coordinate of its center?
The polar equation $r = 6\sin\theta$ is a circle. In rectangular coordinates, it is $x^2 + y^2 = 6y$ . What is the $y$ -coordinate of its center?
The polar equation $r = 8\sin\theta$ is a circle. In rectangular coordinates, it is $x^2 + y^2 = 8y$ . What is the $y$ -coordinate of its center?

The polar equation $\theta =\frac{\pi}{6}$ is a line through the origin. What is its slope?
The polar equation $\theta =\frac{5 \pi}{6}$ is a line through the origin. What is its slope?

Exit ticket

Three questions to check you have got it.

The polar equation $r = 12\sin\theta$ is a circle. In rectangular coordinates, it is $x^2 + y^2 = 12y$ . What is the $y$ -coordinate of its center?
The polar equation $r = 6\sin\theta$ is a circle. In rectangular coordinates, it is $x^2 + y^2 = 6y$ . What is the $y$ -coordinate of its center?
The polar equation $r = 8\sin\theta$ is a circle. In rectangular coordinates, it is $x^2 + y^2 = 8y$ . What is the $y$ -coordinate of its center?

Summary card

Polar equations

Convert with $x = r\cos\theta$ , $y = r\sin\theta$ , $r^2 = x^2 + y^2$ .
$r = a$ : circle; $\theta = \alpha$ : line; $r = 2a\sin\theta$ : circle of radius $|a|$ through the origin.

Watch out: in $r = 2a\sin\theta$ , $2a$ is the diameter.

6. Complex numbers in polar form

Warm-up

Three quick questions from earlier lessons.

Discover

Plot $z = 3 + 4i$ as the point $(3, 4)$ . How far is it from 0?

Key idea

A complex number is a point of the plane, so it has polar coordinates too.

z = a + bi = r(\cos\theta + i\sin\theta)

, with modulus

r = |z| = \sqrt{a^2 + b^2}

and argument

\theta

, where

\tan\theta = \dfrac{b}{a}

in the quadrant of

(a, b)

. Back:

a = r\cos\theta

b = r\sin\theta

Worked example

Write $z = -2 + 2\sqrt{3}\,i$ in polar form.

r = 4

√(4 + 12).

\theta = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}

tan θ = −√3; the point is left and up: quadrant II.

z = 4\left(\cos\dfrac{2\pi}{3} + i\sin\dfrac{2\pi}{3}\right)

Put together.

Your turn, guided

Your turn: write $2(\cos 60^\circ + i\sin 60^\circ)$ as $a + bi$ .

a = 1

2 cos 60° = 1.

z = 1 + \sqrt{3}\,i

2 sin 60° = √3: type z.

Practice ★

Find the exact modulus $|z|$ of $z = 3 - 8i$ .
Find the exact modulus $|z|$ of $z = 2 - 2i$ .
Find the exact modulus $|z|$ of $z = 8 - 9i$ .

Watch out!

Practice ★★

Find the argument of $z = -6 + 1i$ , between $0^\circ$ and $360^\circ$ , to 1 decimal place.
Find the argument of $z = -5 + 1i$ , between $0^\circ$ and $360^\circ$ , to 1 decimal place.

Write $z = 3\left(\cos 120^\circ + i\sin 120^\circ\right)$ in the form $a + bi$ (exact).
Write $z = 2\left(\cos 120^\circ + i\sin 120^\circ\right)$ in the form $a + bi$ (exact).
Write $z = 2\left(\cos 60^\circ + i\sin 60^\circ\right)$ in the form $a + bi$ (exact).

Exit ticket

Three questions to check you have got it.

Write $z = 3\left(\cos 120^\circ + i\sin 120^\circ\right)$ in the form $a + bi$ (exact).
Write $z = 2\left(\cos 120^\circ + i\sin 120^\circ\right)$ in the form $a + bi$ (exact).
Write $z = 2\left(\cos 60^\circ + i\sin 60^\circ\right)$ in the form $a + bi$ (exact).

Summary card

Polar form of a complex number

$z = r(\cos\theta + i\sin\theta)$ .
$r = \sqrt{a^2 + b^2}$ ; $\tan\theta = \dfrac{b}{a}$ , quadrant of $(a, b)$ .
$a = r\cos\theta$ , $b = r\sin\theta$ .

Watch out: plot the point before choosing $\theta$ .

7. Products and powers in polar form (De Moivre)

Warm-up

Three quick questions from earlier lessons.

Discover

Multiplying by $i$ turns a point a quarter turn around 0: $i \cdot 1 = i$ , $i \cdot i = -1$ . What does multiplying by $2i$ do to the length and the angle of $z$ ?

Doubles the length and adds $90^\circ$
Adds 2 to the length and doubles the angle

Key idea

In polar form, products and powers are easy.

r_1(\cos\alpha + i\sin\alpha) \cdot r_2(\cos\beta + i\sin\beta) = r_1 r_2\big(\cos(\alpha + \beta) + i\sin(\alpha + \beta)\big)

: multiply the moduli, add the arguments. De Moivre:

\big(r(\cos\theta + i\sin\theta)\big)^n = r^n(\cos n\theta + i\sin n\theta)

Worked example

Compute $\left(2(\cos 20^\circ + i\sin 20^\circ)\right)^3$ .

2^3 = 8

Modulus to the power 3.

3 \cdot 20^\circ = 60^\circ

Argument times 3.

8(\cos 60^\circ + i\sin 60^\circ) = 4 + 4\sqrt{3}\,i

cos 60° = 1/2, sin 60° = √3/2.

Your turn, guided

Your turn: $3(\cos 50^\circ + i\sin 50^\circ) \cdot 2(\cos 40^\circ + i\sin 40^\circ)$ .

6(\cos 90^\circ + i\sin 90^\circ)

3 · 2 and 50° + 40°.

= 6i

cos 90° = 0, sin 90° = 1: type the result.

Practice ★

$z_1 = 2\left(\cos 40^\circ + i\sin 40^\circ\right)$ and $z_2 = 4\left(\cos 55^\circ + i\sin 55^\circ\right)$ . Find $z_1 z_2$ in polar form.
$z_1 = 3\left(\cos 50^\circ + i\sin 50^\circ\right)$ and $z_2 = 2\left(\cos 55^\circ + i\sin 55^\circ\right)$ . Find $z_1 z_2$ in polar form.
$z_1 = 5\left(\cos 30^\circ + i\sin 30^\circ\right)$ and $z_2 = 6\left(\cos 45^\circ + i\sin 45^\circ\right)$ . Find $z_1 z_2$ in polar form.

Watch out!

Practice ★★

Use De Moivre's theorem to write $\left(\cos 60^\circ + i\sin 60^\circ\right)^{ 2 }$ in the form $a + bi$ (exact).
Use De Moivre's theorem to write $\left(\cos 60^\circ + i\sin 60^\circ\right)^{ 4 }$ in the form $a + bi$ (exact).
Use De Moivre's theorem to write $\left(\cos 30^\circ + i\sin 30^\circ\right)^{ 6 }$ in the form $a + bi$ (exact).

Exit ticket

Three questions to check you have got it.

Use De Moivre's theorem to write $\left(\cos 60^\circ + i\sin 60^\circ\right)^{ 2 }$ in the form $a + bi$ (exact).
Use De Moivre's theorem to write $\left(\cos 60^\circ + i\sin 60^\circ\right)^{ 4 }$ in the form $a + bi$ (exact).
Use De Moivre's theorem to write $\left(\cos 30^\circ + i\sin 30^\circ\right)^{ 6 }$ in the form $a + bi$ (exact).

Summary card

Products and powers in polar form

Product: multiply moduli, add arguments.
De Moivre: $(r\,\mathrm{cis}\,\theta)^n = r^n\,\mathrm{cis}(n\theta)$ .

Watch out: do not raise the real and imaginary parts separately.

8. Distance and midpoint in the complex plane

Warm-up

Three quick questions from earlier lessons.

Discover

The point 3 + 4i is plotted at (3, 4). How far is it from 0?

Key idea

Distance is the modulus of a difference; the midpoint is an average.

Plot

a + bi

(a, b)

. The modulus

|a + bi| = \sqrt{a^2 + b^2}

is its distance from 0. The distance between

z

and

w

|z - w|

, and their midpoint is

\dfrac{z + w}{2}

Worked example

z = 1 + 2i, w = 4 − 2i.

z − w = −3 + 4i

Difference.

|−3 + 4i| = 5

Distance.

(5 + 0i)/2 = 2.5

Midpoint.

Your turn, guided

Your turn: |6 + 8i|.

10

√(36 + 64).

Practice ★

What is the modulus |7 + 8i|, the distance from 7 + 8i to 0?
What is the modulus |2 + 5i|, the distance from 2 + 5i to 0?

What is the real part of the midpoint of 5 + -7i and -2 + -7i?

Watch out!

Practice ★★

What is the distance between 4 + 3i and -4 + -6i in the complex plane?
What is the distance between -6 + 3i and 5 + 0i in the complex plane?

What is the real part of the midpoint of 5 + -7i and -2 + -7i?

Exit ticket

Three questions to check you have got it.

What is the distance between 4 + 3i and -4 + -6i in the complex plane?
What is the distance between -6 + 3i and 5 + 0i in the complex plane?
What is the distance between 3 + 3i and 1 + -3i in the complex plane?

Summary card

Distance and midpoint in the complex plane

|z − w| and (z + w)/2.

Watch out: distance uses √(x² + y²), not x + y.

Answers: Parametric equations and polar coordinates

1. Parametric equations: points and motion

$18$
$17$
$14$

$(2, 1)$
$(1, 0)$

2. Eliminate the parameter

$\left(x - 6\right)^{2}$
$\left(x - 8\right)^{2}$
$\left(x - 5\right)^{2}$

$\frac{4 x}{3} + 9$
$x + 3$
$\frac{4 x}{3} + 7$

A circle of radius 5 centered at the origin

$\frac{4 x}{3} + 9$
$x + 3$
$\frac{4 x}{3} + 7$

3. Circles and ellipses in parametric form

Clockwise

$\frac{\pi}{2}$
$\frac{2 \pi}{3}$
$\pi$

$\frac{\pi}{2}$
$\frac{2 \pi}{3}$
$\pi$

4. Polar coordinates

$\frac{7}{2}$
$- 4 \sqrt{3}$
$\frac{7 \sqrt{3}}{2}$

$\frac{3 \pi}{4}$
$\frac{3 \pi}{4}$

$(0, -4)$
$(-3, 0)$

$\frac{7}{2}$
$- 4 \sqrt{3}$
$\frac{7 \sqrt{3}}{2}$

5. Polar equations and their graphs

A circle of radius 5 centered at the origin
A circle of radius 7 centered at the origin

$\frac{\sqrt{3}}{3}$
$- \frac{\sqrt{3}}{3}$

6. Complex numbers in polar form

$\sqrt{73}$
$2 \sqrt{2}$
$\sqrt{145}$

$- \frac{180 \tan^{-1}{\left(\frac{1}{6} \right)}}{\pi} + 180$
$- \frac{180 \tan^{-1}{\left(\frac{1}{5} \right)}}{\pi} + 180$

$- \frac{3}{2} + \frac{3 \sqrt{3} i}{2}$
$-1 + \sqrt{3} i$
$1 + \sqrt{3} i$

$- \frac{3}{2} + \frac{3 \sqrt{3} i}{2}$
$-1 + \sqrt{3} i$
$1 + \sqrt{3} i$

7. Products and powers in polar form (De Moivre)

$95$
$105$
$75$

$- \frac{1}{2} + \frac{\sqrt{3} i}{2}$
$- \frac{1}{2} - \frac{\sqrt{3} i}{2}$
$-1$

$- \frac{1}{2} + \frac{\sqrt{3} i}{2}$
$- \frac{1}{2} - \frac{\sqrt{3} i}{2}$
$-1$

8. Distance and midpoint in the complex plane

$\sqrt{113}$
$\sqrt{29}$

$\frac{3}{2}$

$\sqrt{145}$
$\sqrt{130}$

$\frac{3}{2}$

$\sqrt{145}$
$\sqrt{130}$
$2 \sqrt{10}$