Step by step
The question
Question: graph and remove t from x = 3 cos t, y = 5 sin t.
1 Make a table
At 0°, 90°, 180°, 270° and 360°, cos and sin are easy: 1, 0 or −1.
They are the x and y of a point going round a circle of radius 1.
Take t = 0°, 90°, 180°, 270° and 360°.
x is 3 times cos t,
and y is 5 times sin t.
2 Plot the points
Draw the grid.
Put each (x, y) on the grid.
At 360° we are back at the start.
3 Join in order of t
Join them in order of t: it goes round, against the clock.
A stretched circle: an ellipse.
4 Remove t
For every t: cos² t + sin² t = 1.
It is Pythagoras on the circle of radius 1: across cos t, up sin t, long side 1.
Get cos t alone: divide x by 3.
And sin t: divide y by 5.
Square both and add them.
Careful!
Not cos t plus sin t: that is not always 1.
Only the squares add up to 1.
That is 1, for every t.
So t is gone.
Square the tops and the bottoms: x² over 9 + y² over 25 = 1.
An ellipse.
5 Check
Let's check with t = 90°: the point (0, 5).
Put it in the answer: we get 1, the right side.
Right!
Let's look back.
First: make a table.
Then: plot the points.
Then: join in order of t.
Then: remove t.
Last: check.