Step by step
The question
Question: write the point (3, −4) in polar form (r, θ).
1 Plot the point
Draw the polar grid, with the x- and y-axes.
Go right 3 and down 4.
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 + 16.
9 plus 16 is 25.
5 times 5 is 25.
So r = 5.
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.
The tan⁻¹ button turns it back into the angle.
The calculator gives about 53.13.
To the nearest tenth, α ≈ 53.1°.
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 down.
θ is a whole turn, 360°, take away α.
So θ = 360° − 53.1°.
Careful!
53.1° alone points right and up.
Our arrow points right and down.
Always look at the picture.
360 minus 53.1 is 306.9.
So θ ≈ 306.9°.
So the polar point is (5, 306.9°).
5 Check
Let's check: go back with x = r cos θ.
The calculator gives 3.
And y = r sin θ gives −4.
We are back at (3, −4).
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.