Sine/cosine parameterisations and circular motion | Year 12

01 / LEARN

The idea, step by step

Repeated paths from sine and cosine

For $x = a cos t, y = b sin t$ , with positive a, b, the Cartesian locus is $x^{2} / a^{2} + y^{2} / b^{2} = 1$ . Equal a, b give a circle; unequal values give an ellipse.

With argument $ω t$ , one full turn takes $2 π / ∣ ω ∣$ . Reversing one sine sign reverses orientation. Add constants to the coordinates to shift the centre.

Circle and helix speeds

A circle of radius R traversed with angular speed magnitude $∣ ω ∣$ has speed $R ∣ ω ∣$ and inward acceleration magnitude $R ω^{2}$ . Relative to the centre, acceleration is $- ω^{2} r$ .

Adding z=ct makes a helix, combining circular planar motion with constant vertical velocity. Its speed is $\sqrt{R^{2} ω^{2} + c^{2}}$ . A turn rises by c times the period.

A common trap

Every sine/cosine ellipse has constant speed.

Instead: Unequal axis lengths generally give variable speed.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

For $(2 cos 3 t, 2 sin 3 t)$ , find radius, period and speed.

Squared distance from origin is 4.
Angular speed is 3.
Multiply radius and angular speed.

Radius 2; period

2 π / 3

; speed 6.

Example 2

Worked example

Find speed and rise per turn for $(3 cos t, 3 sin t, 4 t)$ .

Planar velocity magnitude is 3 and vertical rate 4.
Total magnitude is 5.
A turn lasts $2 π$ , so rise is $4 (2 π)$ .

Speed 5; rise

8 π

Check your understanding

Find period for $(4 cos 2 t, - 4 sin 2 t)$ .

Show answer

$π$

Angular speed magnitude is 2.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF 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 $π / 2$ 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.
$x^{2} / 9 + y^{2} / 4 = 1$ .
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 $π / 2$ 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 topic 6.3–6.5: polynomial and trigonometric parameterisations, related coordinate rates, motion, tangent directions and arc length. This is school Vector calculus, not vector-field calculus.

Official curriculum source ↗

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