Step by step
The question
Question: write the polar point (4, 7π over 4) in rectangular form (x, y).
1 Plot the point
Draw the polar grid: circles 1, 2, 3 around the center, and lines for the angles.
Turn 7π over 4 from the right side of the x-axis, going up and round.
Then go out 4.
π is 180°.
So 7π over 4 is 315°.
2 Use x = r cos θ, y = r sin θ
For any polar point: x = r cos θ and y = r sin θ.
They come from the right triangle under the point.
First x: 4 times cos 315°.
Careful!
x goes with cos, y goes with sin.
Swapped, we would get the point mirrored.
cos 315° is √2/2: the same size as cos 45°, from the special triangles.
It is positive here.
Put it in.
4 times √2/2 is 2√2.
Now y: 4 times sin 315°.
sin 315° is −√2/2: the same size as sin 45°, from the special triangles.
It is negative: the point is below the x-axis.
Put it in.
4 times −√2/2 is −2√2.
So the point is (2√2, −2√2).
3 Check
Let's check on the grid.
From the point, go straight to the x-axis
and straight to the y-axis.
The grid shows x about 2.8 and y about −2.8.
2√2 is about 2.8.
−2√2 is about −2.8.
The same.
Right!
Let's look back.
First: plot the point.
Then: use x = r cos θ, y = r sin θ.
Last: check.