# Sin 210 Degrees - Exact Value −1/2 Explained

## TL;DR  
The value of sin 210 degrees is exactly −\frac{1}{2}, or −0.5. This article finds it with the reference angle of 30°, explains why the sign is negative (210° lies in Quadrant III), gives a standard-angle table, and works through examples and common mistakes.

The value of **sin 210 degrees** is −\frac{1}{2}, or −0.5.  
> **Quick Answer**  
> **Result:** \sin 210°=−\frac{1}{2}  
> **Decimal:** −0.5  
> **In radians:** \sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}  
> **Reference angle:** 30° (since 210°−180°=30°)  
> **Method shown:** reference angle, Quadrant III sign

## Quick Reference Table of Sine Values  
| Angle (degrees) | Angle (radians) | \sin θ (exact) | \sin θ (decimal) |  
| --- | --- | --- | --- |  
| 30° | \frac{\pi}{6} | \frac{1}{2} | 0.5 |  
| 60° | \frac{\pi}{3} | \frac{\sqrt{3}}{2} | 0.866 |  
| 150° | \frac{5\pi}{6} | \frac{1}{2} | 0.5 |  
| 180° | \pi | 0 | 0.0 |  
| 210° | \frac{7\pi}{6} | −\frac{1}{2} | −0.5 |  
| 240° | \frac{4\pi}{3} | −\frac{\sqrt{3}}{2} | −0.866 |  
| 270° | \frac{3\pi}{2} | −1 | −1.0 |

Notice that \sin 210° and \sin 30° share the magnitude \frac{1}{2} but carry opposite signs, because 30° is the **reference angle** of 210° and sine is negative in Quadrant III. Its Quadrant III neighbor \sin 240° uses a 60° reference angle instead.

## What Sin 210 Degrees Means  
On the **unit circle** — a circle of radius 1 centered at the origin — the sine of an angle is the y-coordinate of the point where the angle's radius meets the circle. The angle 210° passes 180° into the lower-left region, **Quadrant III**, landing at \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right), whose negative y-coordinate gives \sin 210° = -\frac{1}{2}.

## How to Find the Value of Sin 210 Degrees

### **Method 1: Reference angle**  
For a Quadrant III angle, the reference angle is:  
210°−180°=30°.  
The sine magnitude matches the reference angle: \sin 30°=\frac{1}{2}. Now fix the sign. The angle 210° is in Quadrant III, where sine is **negative**:  
\sin 210° = -\sin 30° = -\frac{1}{2}.  
**Final answer:** \sin 210°=-\frac{1}{2}=-0.5.

### **Method 2: Unit circle**  
Rotate the unit radius 210° counterclockwise. It lands at \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right), and the sine is the y-coordinate:  
\sin 210°=y\text{-coordinate}=−\frac{1}{2}.

### **Method 3: Supplementary-style identity**  
Writing 210° as 180°+30° and using \sin(180°+θ)=−\sin θ:  
\sin 210°=\sin(180°+30°)=−\sin 30°=−\frac{1}{2}.

## Examples of Sin 210 Degrees  
### Example 1  
Evaluate 6\sin 210°.  
6\sin 210°=6⋅(−\frac{1}{2})=−3.

### Example 2  
Find \sin 210°, but watch which standard angle the magnitude comes from.  
A frequent slip is to reuse the neighbor's reference angle and write \sin 210°=−\sin 60°, mixing 210° up with 240°. Check the subtraction: 210°−180°=30°, not 60°. So the magnitude is \sin 30°=\frac{1}{2}, and:  
\sin 210°=−\sin 30°=−\frac{1}{2}.

### Example 3  
Evaluate \sin 210°+\sin 30°.  
−\frac{1}{2}+\frac{1}{2}=0.

### Example 4  
Verify \sin^2 210°+\cos^2 210°=1, given \cos 210°=−\frac{\sqrt{3}}{2}.  
(−\frac{1}{2})^2+(−\frac{\sqrt{3}}{2})^2=\frac{1}{4} + \frac{3}{4} = 1.

### Example 5  
Express 210° in radians and state the value.  
210°=210×\frac{\pi}{180}=\frac{7\pi}{6}, so \sin\left(\frac{7\pi}{6}\right)=-\frac{1}{2}.

## Common Mistakes With Sin 210 Degrees  
### Mistake 1  
Dropping the negative sign.  
**Where it slips in:** Reading the magnitude \frac{1}{2} off the reference angle and forgetting the Quadrant III sign.  
**Don't do this:** Writing \sin 210°=\frac{1}{2} because \sin 30°=\frac{1}{2}.  
**The correct way:** Sine is negative in Quadrant III, so \sin 210°=-\frac{1}{2}.

### Mistake 2  
Borrowing the reference angle of 240°.  
**Where it slips in:** Using 60° for 210° and writing \sin 210°=−\frac{\sqrt{3}}{2}.  
**The correct way:** 210°−180°=30°, so the magnitude is \frac{1}{2}.

### Mistake 3  
Treating 210° as Quadrant IV.  
**Where it slips in:** Confusing 210° with an angle near 330°, then using the 360°−θ rule.  
**The correct way:** 210° is in Quadrant III, where the rule is θ−180°=30°.

## Key Takeaways  
- **Sin 210 degrees** equals −\frac{1}{2} (or −0.5), an exact value because 210° is a standard angle.  
- The reference angle is 30°, giving the magnitude \frac{1}{2}; Quadrant III makes it negative.  
- In radians, \sin 210°=\sin\left(\frac{7\pi}{6}\right).
- The most common error is reusing the 60° reference angle from 240°, so always recompute θ−180°.