Unit Circle: Step-by-Step Guide
The unit circle is a circle with radius 1 centered at the origin. For any angle theta, measured counterclockwise from the positive x-axis, the point on the circle is (cos(theta), sin(theta)). If you memorize the values for the angles that are multiples of pi/6 and pi/4, you can find the sine and cosine of many angles quickly.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Bring the angle between 0 and 2 pi. Add or subtract 2 pi until the angle is between 0 and 2 pi.\[\theta' = \theta \bmod 2\pi\]
- Find the quadrant. Quadrant I is 0 to pi/2, II is pi/2 to pi, III is pi to 3 pi/2, and IV is 3 pi/2 to 2 pi.\[Q1: [0, \tfrac{\pi}{2}),\; Q2: [\tfrac{\pi}{2}, \pi),\; Q3: [\pi, \tfrac{3\pi}{2}),\; Q4: [\tfrac{3\pi}{2}, 2\pi)\]
- Find the reference angle. The reference angle is the acute angle between the terminal side and the x-axis.\[\theta_{ref} = \begin{cases} \theta & Q1 \\ \pi - \theta & Q2 \\ \theta - \pi & Q3 \\ 2\pi - \theta & Q4 \end{cases}\]
- Look up cosine and sine for the reference angle. Use the standard values for pi/6, pi/4, pi/3, and so on.\[(\cos, \sin):\; \tfrac{\pi}{6}\!\to\!(\tfrac{\sqrt{3}}{2},\tfrac{1}{2}),\; \tfrac{\pi}{4}\!\to\!(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2}),\; \tfrac{\pi}{3}\!\to\!(\tfrac{1}{2},\tfrac{\sqrt{3}}{2})\]
- Put in the signs for the quadrant. Quadrant I: both positive. II: cos negative, sin positive. III: both negative. IV: cos positive, sin negative.\[(\cos\theta, \sin\theta) = (\pm\cos\theta_{ref}, \pm\sin\theta_{ref})\]
Worked example
Find the unit-circle coordinates at theta = 11*pi/6.
Step 1: 11 pi / 6 is already between 0 and 2 pi.
Step 2: 11 pi / 6 is in Quadrant IV.
Step 3: The reference angle is 2 pi - 11 pi / 6 = pi / 6.
Step 4: cos(pi/6) = root 3 / 2, and sin(pi/6) = 1/2.
Step 5: In Quadrant IV, cos is positive and sin is negative. The point is (root 3 / 2, -1/2).
Check: cos^2 + sin^2 = 3/4 + 1/4 = 1. This confirms the point is on the unit circle.
Tips and common mistakes
- Memorize the angles in the first quadrant: 0, pi/6, pi/4, pi/3, and pi/2. All the others come from these.
- A pattern for sine: 0, 1/2, root 2 / 2, root 3 / 2, 1. (Think root 0 / 2, root 1 / 2, root 2 / 2, root 3 / 2, root 4 / 2.)
- Cosine uses the same numbers in reverse: 1, root 3 / 2, root 2 / 2, 1/2, 0.
- A way to remember which are positive in each quadrant: 'All Students Take Calculus'.
- (cos, sin) is the same as (x, y). Do not mix them up.
- Angles that differ by 2 pi land on the same point.
- A reference angle is always between 0 and pi/2.
- Negative angles go clockwise. Add 2 pi if you prefer a positive angle.