Step by step
The question
Question: write the point (−2, 2) in polar form (r, θ).
1 Plot the point
Draw the polar grid, with the x- and y-axes.
Go left 2 and up 2.
Put the point there.
r is the distance from the center to the point.
Across and up make a right triangle.
r is its long side.
2 Find r
r is the distance from the center: the long side of the right triangle.
Pythagoras: square the two parts, add them, then take the square root.
Write the squares: 4 + 4.
4 plus 4 is 8.
But 8 has a square number in it: 4.
8 is 4 times 2.
A root of a times splits into two roots.
√4 is 2.
So r = 2√2.
That is the exact answer, made simple.
3 Find the angle in the triangle
In the right triangle, tan is opposite over adjacent: up over across.
We use the sizes, with no signs.
We call this small angle α.
2 divided by 2 is 1.
The tan⁻¹ button turns it back into the angle.
The calculator gives exactly 45.
So α = 45°.
Here is α on the grid.
4 Which way it points
The direction θ always turns from the right side of the x-axis, going up and round.
The arrow points left and up.
θ is a half turn, 180°, take away α.
So θ = 180° − 45°.
Careful!
45° alone points right and up.
Our arrow points left and up.
Always look at the picture.
180 minus 45 is 135.
So θ = 135°.
So the polar point is (2√2, 135°).
5 Check
Let's check: go back with x = r cos θ.
The calculator gives −2.
And y = r sin θ gives 2.
We are back at (−2, 2).
Right!
Let's look back.
First: plot the point.
Then: find r.
Then: find the angle in the triangle.
Then: which way it points.
Last: check.