Unit Circle | - Free Printable

Let’s solve this step by step.

We are given a unit circle with angles in both degrees and radians, and coordinates (x, y) for each point on the circle. The question is asking us to find the value of sin(210°).

Step 1: Understand what sine means on the unit circle.
On the unit circle, for any angle θ, the sine of that angle is equal to the y-coordinate of the point where the terminal side of the angle intersects the circle.

So, sin(θ) = y

Step 2: Find 210° on the unit circle.
Looking at the diagram, 210° is located in the third quadrant — it’s between 180° and 270°. Specifically, it’s 30° past 180° (since 180° + 30° = 210°).

Step 3: Read the coordinates for 210°.
From the image, the point labeled for 210° has coordinates:
> (-√3/2 , -1/2)

That means:
- x = -√3/2 → this is cos(210°)
- y = -1/2 → this is sin(210°)

Step 4: Therefore, sin(210°) = y = -1/2

Double-check: In the third quadrant, sine is negative (because y-values are below the x-axis), so a negative answer makes sense. Also, 210° is a standard angle — its reference angle is 30°, and sin(30°) = 1/2, so sin(210°) = -sin(30°) = -1/2. That matches.

Final Answer:
-1/2