Step by step
The question
Question: write the point (3, 3) in polar form (r, θ).
1 Plot the point
Draw the polar grid, with the x- and y-axes.
Go right 3 and up 3.
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: 9 + 9.
9 plus 9 is 18.
But 18 has a square number in it: 9.
18 is 9 times 2.
A root of a times splits into two roots.
√9 is 3.
So r = 3√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.
3 divided by 3 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 right and up.
So θ is just α.
So θ = 45°.
So the polar point is (3√2, 45°).
5 Check
Let's check: go back with x = r cos θ.
The calculator gives 3.
And y = r sin θ gives 3.
We are back at (3, 3).
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.