The value of sin 240 degrees is $-\dfrac{\sqrt{3}}{2} \approx -0.8660$.
Quick Answer:
Result: $\sin 240° = -\dfrac{\sqrt{3}}{2}$
Decimal: $-0.8660$ (to four places)
In radians: $\sin\left(\dfrac{4\pi}{3}\right) = -\dfrac{\sqrt{3}}{2}$
Reference angle: $60°$ (since $240° - 180° = 60°$)
Method shown: reference angle, Quadrant III sign
Quick Reference Table of Sine Values
Two hundred forty degrees is a standard angle, so its sine is an exact value. The table runs the related angles across Quadrants I to III, showing how the sign flips once you pass $180°$.
Angle (degrees) | Angle (radians) | $\sin\theta$ (exact) | $\sin\theta$ (decimal) |
|---|---|---|---|
$30°$ | $\dfrac{\pi}{6}$ | $\dfrac{1}{2}$ | $0.5000$ |
$60°$ | $\dfrac{\pi}{3}$ | $\dfrac{\sqrt{3}}{2}$ | $0.8660$ |
$120°$ | $\dfrac{2\pi}{3}$ | $\dfrac{\sqrt{3}}{2}$ | $0.8660$ |
$180°$ | $\pi$ | $0$ | $0.0000$ |
$210°$ | $\dfrac{7\pi}{6}$ | $-\dfrac{1}{2}$ | $-0.5000$ |
$240°$ | $\dfrac{4\pi}{3}$ | $-\dfrac{\sqrt{3}}{2}$ | $-0.8660$ |
$270°$ | $\dfrac{3\pi}{2}$ | $-1$ | $-1.0000$ |
Notice that $\sin 240°$ and $\sin 60°$ share the magnitude $\frac{\sqrt{3}}{2}$ but carry opposite signs, because $60°$ is the reference angle of $240°$ and sine is negative in Quadrant III. The sibling value $\sin 210°$ uses the same logic with a $30°$ reference angle — compare sin 210 degrees.
What Sin 240 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 four quarters of the circle are called quadrants, numbered counterclockwise from Quadrant I (top right). The angle $240°$ sweeps past $180°$ into the lower-left region, Quadrant III, landing at $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$ whose negative $y$-coordinate gives $\sin 240° = -\frac{\sqrt{3}}{2}$.
The right-triangle definition only covers acute angles, so for $240°$ 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 supplies the magnitude, while the quadrant supplies the sign.
How to Find the Value of Sin 240 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 $180°$:
$$240° - 180° = 60°.$$
The sine magnitude matches the reference angle: $\sin 60° = \frac{\sqrt{3}}{2}$. Now fix the sign. The angle $240°$ is in Quadrant III, where sine is negative (only tangent is positive there):
$$\sin 240° = -\sin 60° = -\frac{\sqrt{3}}{2}.$$
Final answer: $\sin 240° = -\dfrac{\sqrt{3}}{2} \approx -0.8660.$
Method 2: Unit circle
Rotate the unit radius $240°$ counterclockwise. It lands at $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$, and the sine is the $y$-coordinate:
$$\sin 240° = y\text{-coordinate} = -\frac{\sqrt{3}}{2}.$$
Method 3: Supplementary-style identity
Writing $240°$ as $180° + 60°$ and using $\sin(180° + \theta) = -\sin\theta$:
$$\sin 240° = \sin(180° + 60°) = -\sin 60° = -\frac{\sqrt{3}}{2}.$$
Examples of Sin 240 Degrees
Example 1: Evaluate $2\sin 240°$
$$2\sin 240° = 2\cdot\left(-\frac{\sqrt{3}}{2}\right) = -\sqrt{3} \approx -1.732.$$
Example 2: Find $\sin 240°$, but watch the reference angle
The quick instinct is to subtract from $360°$: $360° - 240° = 120°$, then call $120°$ the reference angle. That is wrong, because $120°$ is not acute, and a reference angle must be between $0°$ and $90°$. For a Quadrant III angle you subtract $180°$, not $360°$:
$$240° - 180° = 60°.$$
So $\sin 240° = -\sin 60° = -\frac{\sqrt{3}}{2}$. The $360° - \theta$ rule belongs to Quadrant IV, not Quadrant III.
Example 3: Evaluate $\sin 240° + \sin 60°$
$$-\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = 0.$$
The reference-angle partners cancel — same magnitude, opposite sign across the half-turn.
Example 4: Verify $\sin^2 240° + \cos^2 240° = 1$, given $\cos 240° = -\dfrac{1}{2}$
$$\left(-\frac{\sqrt{3}}{2}\right)^2 + \left(-\frac{1}{2}\right)^2 = \frac{3}{4} + \frac{1}{4} = 1.$$
The Pythagorean identity holds even with both functions negative — squaring removes the signs.
Example 5
Express $240°$ in radians and state the value.
$240° = 240 \times \frac{\pi}{180} = \frac{4\pi}{3}$ radians, so $\sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}$. Converting between radians and degrees does not change the value.
Common Mistakes With Sin 240 Degrees
Mistake 1: Dropping the negative sign
Where it slips in: Reading the magnitude $\frac{\sqrt{3}}{2}$ off the reference angle and forgetting the Quadrant III sign.
Don't do this: Writing $\sin 240° = \frac{\sqrt{3}}{2}$ because $\sin 60° = \frac{\sqrt{3}}{2}$.
The correct way: Sine is negative in Quadrant III, so $\sin 240° = -\frac{\sqrt{3}}{2}$. The habit that fixes it is to mark the quadrant before writing the magnitude — the learner who skips the quadrant check is the one who loses the sign.
Mistake 2: Using the wrong reference-angle rule
Where it slips in: Subtracting from $360°$ instead of $180°$ for a Quadrant III angle.
Don't do this: Calling the reference angle $360° - 240° = 120°$.
The correct way: Quadrant III uses $\theta - 180°$, giving $60°$. The $360° - \theta$ rule is for Quadrant IV. A reference angle is always acute, so $120°$ should look wrong on sight.
Mistake 3: Confusing the reference angles of 240° and 210°
Where it slips in: Both angles sit in Quadrant III, so students assume they share a reference angle.
Don't do this: Using $60°$ for $\sin 210°$ or $30°$ for $\sin 240°$.
The correct way: $240° - 180° = 60°$, but $210° - 180° = 30°$. They land in the same quadrant with different reference angles, so $\sin 240° = -\frac{\sqrt{3}}{2}$ while $\sin 210° = -\frac{1}{2}$.
Key Takeaways
Sin 240 degrees equals $-\frac{\sqrt{3}}{2}$ (about $-0.8660$), an exact value because $240°$ is a standard angle.
The reference angle is $60°$, giving the magnitude $\frac{\sqrt{3}}{2}$; Quadrant III makes it negative.
In radians, $\sin 240° = \sin\left(\frac{4\pi}{3}\right)$.
The most common error is dropping the negative sign — always read the quadrant before the magnitude.
Practice These Before Moving On
Find the reference angle of $240°$ and use it to write $\sin 240°$ from scratch.
Evaluate $4\sin 240° + 2\cos 240°$ using $\cos 240° = -\frac{1}{2}$.
Convert $240°$ to radians and write the value as $\sin\left(\frac{4\pi}{3}\right)$.
To get the reference-angle and quadrant-sign method drilled with a teacher, Bhanzu's trigonometry tutor and high school math tutor sessions work straight off the unit circle, with math classes online for live practice.
Read More
Cos 120 degrees — a Quadrant II value using the same reference-angle logic.
Cos 135 degrees — another Quadrant II value via reference angle.
Cos 270 degrees — an axis value at the bottom of the circle.
Trigonometric ratios — how sine, cosine, and tangent are defined.
Trigonometric table — every standard-angle value in one chart.
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