Polar Coordinates: Step-by-Step Guide
Polar coordinates describe a point using its distance r from the origin and its angle theta from the positive x-axis. To change to regular (x, y) coordinates, use x = r cos(theta) and y = r sin(theta). To go the other way, use r = the square root of (x^2 + y^2) and theta = atan2(y, x).
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Polar to rectangular. Given (r, theta), find x = r cos(theta) and y = r sin(theta).\[x = r\cos\theta,\quad y = r\sin\theta\]
- Rectangular to polar: find r. Given (x, y), find r = the square root of (x^2 + y^2). We usually keep r positive.\[r = \sqrt{x^{2}+y^{2}}\]
- Find the angle in the right quadrant. Use theta = atan2(y, x), or find the reference angle arctan(|y/x|) and adjust for the signs of x and y.\[\theta = \operatorname{atan2}(y, x)\]
- Write the answer and check. Write it as (r, theta) or (x, y), and check by converting back the other way.\[(r,\theta) \leftrightarrow (x,y)\]
Worked example
Convert the polar point (4, 5 pi/6) to rectangular coordinates.
Step 1: r = 4 and theta = 5 pi / 6. Use x = r cos(theta) and y = r sin(theta).
Step 2: cos(5 pi / 6) = -root 3 / 2 and sin(5 pi / 6) = 1/2.
Step 3: x = 4 x (-root 3 / 2) = -2 root 3.
Step 4: y = 4 x (1/2) = 2.
Step 5: The point in rectangular coordinates is (-2 root 3, 2).
Check: r = the square root of ((-2 root 3)^2 + 2^2) = the square root of (12 + 4) = 4. The point is in Quadrant II, and the reference angle is pi/6, so the angle is pi - pi/6 = 5 pi / 6. It matches.
Tips and common mistakes
- A polar point can be written many ways. (r, theta) is the same as (r, theta + 2 pi k), and (-r, theta) is the same as (r, theta + pi).
- Usually r is kept positive and theta is between 0 and 2 pi (or between -pi and pi).
- When you go from (x, y) to polar, check the quadrant. Arctan alone can give the wrong angle.
- If x = 0, theta is pi/2 (if y > 0) or 3 pi / 2 (if y < 0). If y = 0 and x < 0, theta = pi.
- Memorize cosine and sine of 0, pi/6, pi/4, pi/3, and pi/2 and the matching angles in other quadrants.
- On a calculator, use atan2(y, x) instead of arctan(y/x). It gets the quadrant right for you.
- Sketch the point first. It tells you the quadrant and a rough angle, which helps you catch sign mistakes.