Sin 210 Degrees - Exact Value −1/2 Explained

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 $180°$.

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

1. Find the reference angle of $210°$ and use it to write $\sin 210°$ from scratch.

2. Evaluate $2\sin 210° + 4\cos 210°$ using $\cos 210° = -\frac{\sqrt{3}}{2}$.

3. 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.