The value of sin 210 degrees is $-\dfrac{1}{2}$, or $-0.5$.
Quick Answer
Result: $\sin 210° = -\dfrac{1}{2}$
Decimal: $-0.5$
In radians: $\sin\left(\dfrac{7\pi}{6}\right) = -\dfrac{1}{2}$
Reference angle: $30°$ (since $210° - 180° = 30°$)
Method shown: reference angle, Quadrant III sign
Quick Reference Table of Sine Values
Two hundred ten degrees is a standard angle, so its sine is exact. The table runs the related angles across Quadrants I to III, showing the sign flip past $180°$.
Angle (degrees) | Angle (radians) | $\sin\theta$ (exact) | $\sin\theta$ (decimal) |
|---|---|---|---|
$30°$ | $\dfrac{\pi}{6}$ | $\dfrac{1}{2}$ | $0.5000$ |
$60°$ | $\dfrac{\pi}{3}$ | $\dfrac{\sqrt{3}}{2}$ | $0.8660$ |
$150°$ | $\dfrac{5\pi}{6}$ | $\dfrac{1}{2}$ | $0.5000$ |
$180°$ | $\pi$ | $0$ | $0.0000$ |
$210°$ | $\dfrac{7\pi}{6}$ | $-\dfrac{1}{2}$ | $-0.5000$ |
$240°$ | $\dfrac{4\pi}{3}$ | $-\dfrac{\sqrt{3}}{2}$ | $-0.8660$ |
$270°$ | $\dfrac{3\pi}{2}$ | $-1$ | $-1.0000$ |
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 neighbour $\sin 240°$ uses a $60°$ reference angle instead — compare sin 240 degrees.
What Sin 210 Degrees Means
On the unit circle — a circle of radius $1$ centred at the origin — the sine of an angle is the $y$-coordinate of the point where the angle's radius meets the circle. The circle's four quarters are called quadrants, numbered counterclockwise from Quadrant I (top right). 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 $\sin 210° = -\frac{1}{2}$.
The right-triangle definition only covers acute angles, so for $210°$ the unit circle is the home definition. The triangle still helps through the reference angle — the acute angle between the radius and the $x$-axis — which gives the magnitude, while the quadrant gives the sign.
How to Find the Value of Sin 210 Degrees
The reference-angle method does it in two moves: read the size from the acute partner, then fix the sign from the quadrant.
Method 1: Reference angle
For a Quadrant III angle, the reference angle is the angle past $180°$:
$$210° - 180° = 30°.$$
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: $\sin 210° = -\dfrac{1}{2} = -0.5.$
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 $\sin(180° + \theta) = -\sin\theta$:
$$\sin 210° = \sin(180° + 30°) = -\sin 30° = -\frac{1}{2}.$$
Examples of Sin 210 Degrees
Example 1
Evaluate $6\sin 210°$.
$$6\sin 210° = 6\cdot\left(-\frac{1}{2}\right) = -3.$$
Example 2
Find $\sin 210°$, but watch which standard angle the magnitude comes from.
A frequent slip is to reuse the neighbour's reference angle and write $\sin 210° = -\sin 60° = -\frac{\sqrt{3}}{2}$, mixing $210°$ up with $240°$. Check the subtraction: $210° - 180° = 30°$, not $60°$. So the magnitude is $\sin 30° = \frac{1}{2}$, and:
$$\sin 210° = -\sin 30° = -\frac{1}{2}.$$
The reference angle has to be computed for this angle, not borrowed from the one beside it.
Example 3
Evaluate $\sin 210° + \sin 30°$.
$$-\frac{1}{2} + \frac{1}{2} = 0.$$
The reference-angle partners cancel — same magnitude, opposite sign across the half-turn.
Example 4
Verify $\sin^2 210° + \cos^2 210° = 1$, given $\cos 210° = -\dfrac{\sqrt{3}}{2}$.
$$\left(-\frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1.$$
The Pythagorean identity holds with both functions negative — squaring removes the signs.
Example 5
Express $210°$ in radians and state the value.
$210° = 210 \times \frac{\pi}{180} = \frac{7\pi}{6}$ radians, so $\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}$. Converting between radians and degrees leaves the value unchanged.
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 $\sin 210° = -\frac{1}{2}$. The habit that fixes it is to mark the quadrant before writing the magnitude — the sign comes from where the angle lands, not from the acute partner.
Mistake 2: Borrowing the reference angle of 240°
Where it slips in: Both $210°$ and $240°$ sit in Quadrant III, so the reference angles get swapped.
Don't do this: Using $60°$ for $210°$ and writing $\sin 210° = -\frac{\sqrt{3}}{2}$.
The correct way: $210° - 180° = 30°$, so the magnitude is $\frac{1}{2}$. The learner who memorises one Quadrant III value and reuses it on the next is the one this catches — always recompute $\theta - 180°$.
Mistake 3: Treating 210° as Quadrant IV
Where it slips in: Confusing $210°$ (just past $180°$) with an angle near $330°$, then using the $360° - \theta$ rule.
Don't do this: Calling the reference angle $360° - 210° = 150°$.
The correct way: $210°$ is in Quadrant III, where the rule is $\theta - 180° = 30°$. A reference angle is always acute, so $150°$ should look wrong immediately.
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 $\theta - 180°$.
Practice These Before Moving On
Find the reference angle of $210°$ and use it to write $\sin 210°$ from scratch.
Evaluate $2\sin 210° + 4\cos 210°$ using $\cos 210° = -\frac{\sqrt{3}}{2}$.
Convert $210°$ to radians and write the value as $\sin\left(\frac{7\pi}{6}\right)$.
To make the reference-angle and quadrant-sign routine automatic with a teacher, Bhanzu's trigonometry tutor and high school math tutor sessions work straight from the unit circle, with math classes online for live practice.
Read More
Cos 135 degrees — a Quadrant II value using the same reference-angle method.
Cos 120 degrees — another Quadrant II value via reference angle.
Cos 270 degrees — an axis value at the bottom of the circle.
Sin Cos Tan — the three core ratios and how they connect.
Trigonometric table — every standard-angle value in one chart.
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