Sine/cosine parameterisations and circular motion | Year 12

01 / LEARN

Understand the method.

THE CORE RULE

\begin{aligned} r (t) & = (\begin{array}{cccccccccc} h + R cos ω t \\ k + R sin ω t \end{array}) \\ T & = \frac{2 π}{∣ ω ∣} \\ s & = R ∣ ω ∣ \end{aligned}

For $R > 0$ and nonzero constant $ω$ , the path is a radius- $R$ circle centred at $(h, k)$ . One revolution takes $T$ time units.

Separate the path from its timing

For $x = h + a cos ω t$ , $y = k + b sin ω t$ , with $a, b > 0$ and $ω \neq 0$ , subtract the centre coordinates, divide by the amplitudes, square and add. The result is an ellipse; equal amplitudes give a circle.

The Cartesian equation identifies the path, but it does not say how quickly or in which direction the object moves. Use the parameter functions for that information.

\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1

Read the angular rate and the motion

One turn changes the angle by $2 π$ . With constant nonzero angular rate $ω$ , the period is $2 π / ∣ ω ∣$ . Check the starting point and velocity to establish direction; changing one sine sign reverses the orientation.

On a circle, speed is $R ∣ ω ∣$ and acceleration points inward, with magnitude $R ω^{2}$ . A shifted circle uses position relative to its centre $c = (h, k)$ .

\begin{aligned} a & = - ω^{2} (r - c) \\ ∣ a ∣ & = R ω^{2} \end{aligned}

Check speed instead of assuming it

An ellipse with unequal amplitudes generally has variable speed: its squared velocity components do not combine to one constant. Differentiate and calculate the magnitude.

A helix adds a constant vertical rate: $r (t) = (R cos ω t, R sin ω t, c t)$ . Its horizontal and vertical velocity components are perpendicular, so combine their magnitudes using Pythagoras. Its vertical change per turn is $c T$ , which can be positive or negative.

\begin{aligned} s & = \sqrt{R^{2} ω^{2} + c^{2}} \\ Δ z & = c \frac{2 π}{∣ ω ∣} \end{aligned}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Describe a circular route

A particle has position $r (t) = (2 cos 3 t, 2 sin 3 t)$ metres, with $t$ in seconds. Find its radius, period, speed and acceleration magnitude. State its direction from $t = 0$ .

A radius-2 circle centred at the origin, starting at (2,0) and traversed counterclockwise, with tangent velocity upward and acceleration directed inward.

Worked solution

1
Identify the radius and angular rate.
$\begin{array}{c} x^{2} + y^{2} \\ = 4 ({cos}^{2} 3 t + {sin}^{2} 3 t) \\ = 4 \\ R = 2, ω = 3 \end{array}$
Equal amplitudes give a circle centred at the origin. The angle increases by $3$ radians each second.
2
Find the time for one revolution.
$\begin{aligned} 3 T & = 2 π \\ T & = \frac{2 π}{3} \end{aligned}$ s.
A complete revolution corresponds to an angular increase of $2 π$ .
3
Differentiate and calculate the magnitudes.
$\begin{aligned} v (t) & = (\begin{array}{cccccccccc} - 6 sin 3 t \\ 6 cos 3 t \end{array}) \\ s & = \sqrt{\begin{array}{c} 36 {sin}^{2} 3 t \\ + 36 {cos}^{2} 3 t \end{array}} \\ = 6 \\ a (t) & = (\begin{array}{cccccccccc} - 18 cos 3 t \\ - 18 sin 3 t \end{array}) \\ ∣ a ∣ & = 18 \end{aligned}$
The trigonometric identity makes both magnitudes constant. Speed is in m/s and acceleration magnitude is in m/s².
4
Inspect the starting point and velocity.
$\begin{aligned} r (0) & = (2, 0) \\ v (0) & = (0, 6) \end{aligned}$
At the rightmost point, upward velocity means counterclockwise motion. The acceleration $(- 18, 0)$ points toward the centre.

Answer

Radius $2$ m, period $\frac{2 π}{3}$ s, speed $6$ m/s, and acceleration magnitude $18$ m/s². Motion is counterclockwise from $(2, 0)$ .

EXAMPLE 02

Test the speed around an ellipse

For $r (t) = (4 cos t, 2 sin t)$ metres, find the Cartesian path and compare the speeds at $t = 0$ and $t = \frac{π}{2}$ seconds. Is the speed constant?

Worked solution

1
Divide by the amplitudes, square and add.
$\begin{array}{c} \frac{x^{2}}{16} + \frac{y^{2}}{4} \\ = {cos}^{2} t + {sin}^{2} t \\ = 1 \end{array}$
The unequal semiaxes $4$ m and $2$ m give an ellipse.
2
Differentiate before taking magnitudes.
$\begin{aligned} v (t) & = (\begin{array}{cccccccccc} - 4 sin t \\ 2 cos t \end{array}) \\ s (t) & = \sqrt{\begin{array}{c} 16 {sin}^{2} t \\ + 4 {cos}^{2} t \end{array}} \end{aligned}$
The coefficients $16$ and $4$ differ, so the trigonometric identity does not turn this expression into one constant.
3
Compare two times on the same route.
$\begin{aligned} v (0) & = (0, 2) \\ s (0) & = 2 \\ v (\frac{π}{2}) & = (- 4, 0) \\ s (\frac{π}{2}) & = 4 \end{aligned}$
Two different speed values prove that speed is not constant, even though the angular parameter increases uniformly.

Answer

The path is $\frac{x^{2}}{16} + \frac{y^{2}}{4} = 1$ . The speeds are $2$ m/s and $4$ m/s, so the speed is not constant.

EXAMPLE 03

Combine rotation with a steady climb

An inspection drone follows $r (t) = (3 cos t, 3 sin t, 4 t)$ metres for one revolution, starting at $t = 0$ . Find its speed, the time for one turn, and its rise, with $t$ in seconds.

Worked solution

1
Differentiate all three coordinates.
$v (t) = (\begin{array}{cccccccccc} - 3 sin t \\ 3 cos t \\ 4 \end{array})$
The drone combines circular horizontal motion with a constant vertical velocity of $4$ m/s.
2
Combine the squared velocity components.
$\begin{aligned} s & = \sqrt{\begin{array}{c} 9 {sin}^{2} t + 9 {cos}^{2} t \\ + 16 \end{array}} \\ = \sqrt{9 + 16} \\ = 5 \end{aligned}$
The horizontal speed is $3$ m/s. The perpendicular vertical rate is $4$ m/s, giving total speed $5$ m/s.
3
Use the horizontal angle to time one turn.
$T = 2 π$ s, $\begin{aligned} Δ z & = 4 T \\ = 8 π \end{aligned}$ m.
The horizontal angle is $t$ , so one revolution lasts $2 π$ seconds. The height increases throughout that time.

Answer

Speed $5$ m/s; one turn takes $2 π$ s and rises $8 π$ m. The horizontal coordinates return to their start, but the drone is higher.

03 / YOUR TURN

Try it yourself.

Calculate, explain, then apply.

CHECK 01

Read a shifted circular model

For $r (t) = (1 + 4 cos 2 t, - 2 + 4 sin 2 t)$ metres, find its centre, radius, period, speed and acceleration magnitude. Time is in seconds.

Show answer & reasoning

Centre $(1, - 2)$ m; radius $4$ m; period $π$ s; speed $8$ m/s; acceleration magnitude $16$ m/s².

The constants shift the centre and do not change velocity. With $R = 4$ and $∣ ω ∣ = 2$ , use $T = \frac{2 π}{2}$ , $s = 4 (2)$ and $∣ a ∣ = 4 (2^{2})$ .

CHECK 02

Use velocity to decide direction

A particle follows $r (t) = (2 cos t, - 2 sin t)$ metres. At $t = 0$ , does it move clockwise or counterclockwise? Does changing the sine sign change its speed?

Show answer & reasoning

It moves clockwise, initially downward from $(2, 0)$ . Its speed remains $2$ m/s.

The derivative is $(- 2 sin t, - 2 cos t)$ , giving $(0, - 2)$ at zero. This is downward at the rightmost point. Squaring the components removes the changed sign, leaving speed $2$ .

CHECK 03

Plan a rising scanner’s first turn

A scanner moves along $r (t) = (2 cos 2 t, 2 sin 2 t, 3 t)$ metres. How long does its first full turn take, how far does it rise, and what is its speed? Time is in seconds.

Show answer & reasoning

One turn takes $π$ s, rises $3 π$ m, and has speed $5$ m/s.

The angle is $2 t$ , so $2 T = 2 π$ gives $T = π$ . The height change is $3 T = 3 π$ . Velocity is $(- 4 sin 2 t, 4 cos 2 t, 3)$ , so speed is $\sqrt{16 + 9} = 5$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Find the Cartesian locus of $(3 cos t, 2 sin t)$ .
2 marks
Find speed and acceleration magnitude for $(5 cos 2 t, 5 sin 2 t)$ .
2 marks
At $0$ , describe direction for $(2 cos t, - 2 sin t)$ .
2 marks
Find centre and radius of $(1 + 4 cos t, - 2 + 4 sin t)$ .
2 marks
Find speed for $(2 cos t, 2 sin t, t)$ .
2 marks
Compare speeds at $0$ and $\frac{π}{2}$ for $(4 cos t, 2 sin t)$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the Cartesian locus of $(3 cos t, 2 sin t)$ .
2 marks
Divide by the amplitudes, square and add.
Use the Pythagorean trig identity.
$\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ .
Find speed and acceleration magnitude for $(5 cos 2 t, 5 sin 2 t)$ .
2 marks
Use radius times angular speed for speed.
Use radius times angular speed squared for acceleration.
$10$ and $20$ .
At $0$ , describe direction for $(2 cos t, - 2 sin t)$ .
2 marks
The derivative at zero is $(0, - 2)$ .
That is downward at the rightmost point.
Down from $(2, 0)$ , clockwise.
Find centre and radius of $(1 + 4 cos t, - 2 + 4 sin t)$ .
2 marks
The constants shift the centre.
The equal amplitudes give the radius.
Centre $(1, - 2)$ , radius $4$ .
Find speed for $(2 cos t, 2 sin t, t)$ .
2 marks
Velocity is $(- 2 sin t, 2 cos t, 1)$ .
Squared speed is $4 + 1$ .
$\sqrt{5}$
Compare speeds at $0$ and $\frac{π}{2}$ for $(4 cos t, 2 sin t)$ .
2 marks
Velocity is $(- 4 sin t, 2 cos t)$ .
The specified times give $(0, 2)$ and $(- 4, 0)$ .
$2$ and $4$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics school Vector calculus: polynomial and trigonometric parameterisations, related coordinate rates, Bézier paths, componentwise differentiation and integration, motion, tangent directions, circular motion and arc length. Reasoning and proofs use these school topics; vector-field calculus is outside this chapter.

Official curriculum source ↗

In this lesson

Understand the method.

Separate the path from its timing

Read the angular rate and the motion

Check speed instead of assuming it

See the method in action.

Describe a circular route

Identify the radius and angular rate.

Find the time for one revolution.

Differentiate and calculate the magnitudes.

Inspect the starting point and velocity.

Test the speed around an ellipse

Divide by the amplitudes, square and add.

Differentiate before taking magnitudes.

Compare two times on the same route.

Combine rotation with a steady climb

Differentiate all three coordinates.

Combine the squared velocity components.

Use the horizontal angle to time one turn.

Try it yourself.

Read a shifted circular model

Use velocity to decide direction

Plan a rising scanner’s first turn

Keep practising