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

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°.