Polar Coordinates — Practice
Distance and direction describe a point just as well as across and up.
What these questions cover
These questions use 7 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Radius — Centre to edge. Half the diameter.
- Parabola — The U-shaped graph of a quadratic function.
- Ray — One endpoint and everything beyond it in one direction. Written AB.
- Complex number — Anything of the form a + bi, with a the real part and b the imaginary part.
- Complex plane — The plane where the horizontal axis is real and the vertical axis imaginary.
- Rectangular form — The ordinary (x, y) description.
- Quadrant — One of the four regions the axes divide the plane into.
Problem 1
Find r for the point (3, 4).
Worked answer
5
√9+16 = 5
Problem 2
Find r for the point (-6, 8).
Worked answer
10
√36+64 = 10
Problem 3
Convert (4, 0°) to rectangular form. Give x.
Worked answer
4
x = 4cos 0° = 4
Problem 4
Convert (5, 90°) to rectangular form. Give y.
Worked answer
5
y = 5sin 90° = 5
Problem 5
What is the graph of r = 7?
- A circle of radius 7
- A line through the origin
- A spiral
- A point
Worked answer
A circle of radius 7
Every point is 7 from the origin, in all directions
Problem 6
Convert (6, 60°) to rectangular form. Give x.
Worked answer
3
6cos 60° = 6(0.5) = 3
Problem 7
Convert (8, 30°) to rectangular form. Give y.
Worked answer
4
8sin 30° = 8(0.5) = 4
Problem 8
Find θ in degrees for the point (0, 5), with 0° ≤ θ < 360°.
Worked answer
90
No horizontal part, so the direction is 90°
Problem 9
Find θ in degrees for the point (-3, 0), with 0° ≤ θ < 360°.
Worked answer
180
Pointing along the negative x-axis
Problem 10
What is the graph of θ = 45°?
- A line through the origin
- A circle
- A ray of length 45
- A parabola
Worked answer
A line through the origin
Every distance in that one direction, including negative r backwards
Problem 11
Find r for the point (-5, -12).
Worked answer
13
√25+144 = 13
Problem 12
For the point (-4, 4), find θ in degrees with 0° ≤ θ < 360°.
Worked answer
135
tanθ = -1 gives -45°
The point is up and left, so add 180°: 135°
Problem 13
For the point (-2, -2), find θ in degrees with 0° ≤ θ < 360°.
Worked answer
225
tanθ = 1 gives 45°
Quadrant III, so add 180°: 225°
Problem 14
Give an angle between 0° and 360° naming the same point as (4, 400°).
Worked answer
40
400-360 = 40
Problem 15
The point (-3, 50°) is the same as (3, θ). Find θ in degrees.
Worked answer
230
Reverse the direction: 50+180 = 230
Problem 16
Find |z| for the complex number z = 8 - 6i.
Worked answer
10
√64+36 = 10