Sin 240 Degrees - Exact Value −√3/2 Explained

#Trigonometry
TL;DR
The value of sin 240 degrees is exactly −32-\frac{\sqrt{3}}{2}, about −0.8660. This article finds it with the reference angle of 60°, explains why the sign is negative (240° lies in Quadrant III), gives a standard-angle table, and works through examples and common mistakes.
BT
Bhanzu TeamLast updated on July 16, 20266 min read

The value of sin 240 degrees is −32≈−0.8660-\dfrac{\sqrt{3}}{2} \approx -0.8660.

Quick Answer:

Result: sin⁡240°=−32\sin 240° = -\dfrac{\sqrt{3}}{2}

Decimal: −0.8660-0.8660 (to four places)

In radians: sin⁡(4π3)=−32\sin\left(\dfrac{4\pi}{3}\right) = -\dfrac{\sqrt{3}}{2}

Reference angle: 60°60° (since 240°−180°=60°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°180°.

Angle (degrees)

Angle (radians)

sin⁡θ\sin\theta (exact)

sin⁡θ\sin\theta (decimal)

30°30°

π6\dfrac{\pi}{6}

12\dfrac{1}{2}

0.50000.5000

60°60°

π3\dfrac{\pi}{3}

32\dfrac{\sqrt{3}}{2}

0.86600.8660

120°120°

2π3\dfrac{2\pi}{3}

32\dfrac{\sqrt{3}}{2}

0.86600.8660

180°180°

π\pi

00

0.00000.0000

210°210°

7π6\dfrac{7\pi}{6}

−12-\dfrac{1}{2}

−0.5000-0.5000

240°240°

4π3\dfrac{4\pi}{3}

−32-\dfrac{\sqrt{3}}{2}

−0.8660-0.8660

270°270°

3π2\dfrac{3\pi}{2}

−1-1

−1.0000-1.0000

Notice that sin⁡240°\sin 240° and sin⁡60°\sin 60° share the magnitude 32\frac{\sqrt{3}}{2} but carry opposite signs, because 60°60° is the reference angle of 240°240° and sine is negative in Quadrant III. The sibling value sin⁡210°\sin 210° uses the same logic with a 30°30° reference angle — compare sin 210 degrees.

What Sin 240 Degrees Means

On the unit circle — a circle of radius 11 centred at the origin — the sine of an angle is the yy-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°240° sweeps past 180°180° into the lower-left region, Quadrant III, landing at (−12,−32)\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) whose negative yy-coordinate gives sin⁡240°=−32\sin 240° = -\frac{\sqrt{3}}{2}.

The right-triangle definition only covers acute angles, so for 240°240° the unit circle is the home definition. The triangle still helps through the reference angle — the acute angle between the radius and the xx-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°180°:

240°−180°=60°.240° - 180° = 60°.

The sine magnitude matches the reference angle: sin⁡60°=32\sin 60° = \frac{\sqrt{3}}{2}. Now fix the sign. The angle 240°240° is in Quadrant III, where sine is negative (only tangent is positive there):

sin⁡240°=−sin⁡60°=−32.\sin 240° = -\sin 60° = -\frac{\sqrt{3}}{2}.

Final answer: sin⁡240°=−32≈−0.8660.\sin 240° = -\dfrac{\sqrt{3}}{2} \approx -0.8660.

Method 2: Unit circle

Rotate the unit radius 240°240° counterclockwise. It lands at (−12,−32)\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right), and the sine is the yy-coordinate:

sin⁡240°=y-coordinate=−32.\sin 240° = y\text{-coordinate} = -\frac{\sqrt{3}}{2}.

Method 3: Supplementary-style identity

Writing 240°240° as 180°+60°180° + 60° and using sin⁡(180°+θ)=−sin⁡θ\sin(180° + \theta) = -\sin\theta:

sin⁡240°=sin⁡(180°+60°)=−sin⁡60°=−32.\sin 240° = \sin(180° + 60°) = -\sin 60° = -\frac{\sqrt{3}}{2}.

Examples of Sin 240 Degrees

Example 1: Evaluate 2sin⁡240°2\sin 240°

2sin⁡240°=2⋅(−32)=−3≈−1.732.2\sin 240° = 2\cdot\left(-\frac{\sqrt{3}}{2}\right) = -\sqrt{3} \approx -1.732.

Example 2: Find sin⁡240°\sin 240°, but watch the reference angle

The quick instinct is to subtract from 360°360°: 360°−240°=120°360° - 240° = 120°, then call 120°120° the reference angle. That is wrong, because 120°120° is not acute, and a reference angle must be between 0°0° and 90°90°. For a Quadrant III angle you subtract 180°180°, not 360°360°:

240°−180°=60°.240° - 180° = 60°.

So sin⁡240°=−sin⁡60°=−32\sin 240° = -\sin 60° = -\frac{\sqrt{3}}{2}. The 360°−θ360° - \theta rule belongs to Quadrant IV, not Quadrant III.

Example 3: Evaluate sin⁡240°+sin⁡60°\sin 240° + \sin 60°

−32+32=0.-\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⁡2240°+cos⁡2240°=1\sin^2 240° + \cos^2 240° = 1, given cos⁡240°=−12\cos 240° = -\dfrac{1}{2}

(−32)2+(−12)2=34+14=1.\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°240° in radians and state the value.

240°=240×π180=4π3240° = 240 \times \frac{\pi}{180} = \frac{4\pi}{3} radians, so sin⁡(4π3)=−32\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 32\frac{\sqrt{3}}{2} off the reference angle and forgetting the Quadrant III sign.

Don't do this: Writing sin⁡240°=32\sin 240° = \frac{\sqrt{3}}{2} because sin⁡60°=32\sin 60° = \frac{\sqrt{3}}{2}.

The correct way: Sine is negative in Quadrant III, so sin⁡240°=−32\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°360° instead of 180°180° for a Quadrant III angle.

Don't do this: Calling the reference angle 360°−240°=120°360° - 240° = 120°.

The correct way: Quadrant III uses θ−180°\theta - 180°, giving 60°60°. The 360°−θ360° - \theta rule is for Quadrant IV. A reference angle is always acute, so 120°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°60° for sin⁡210°\sin 210° or 30°30° for sin⁡240°\sin 240°.

The correct way: 240°−180°=60°240° - 180° = 60°, but 210°−180°=30°210° - 180° = 30°. They land in the same quadrant with different reference angles, so sin⁡240°=−32\sin 240° = -\frac{\sqrt{3}}{2} while sin⁡210°=−12\sin 210° = -\frac{1}{2}.

Key Takeaways

  • Sin 240 degrees equals −32-\frac{\sqrt{3}}{2} (about −0.8660-0.8660), an exact value because 240°240° is a standard angle.

  • The reference angle is 60°60°, giving the magnitude 32\frac{\sqrt{3}}{2}; Quadrant III makes it negative.

  • In radians, sin⁡240°=sin⁡(4π3)\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

  1. Find the reference angle of 240°240° and use it to write sin⁡240°\sin 240° from scratch.

  2. Evaluate 4sin⁡240°+2cos⁡240°4\sin 240° + 2\cos 240° using cos⁡240°=−12\cos 240° = -\frac{1}{2}.

  3. Convert 240°240° to radians and write the value as sin⁡(4π3)\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.

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Frequently Asked Questions

What is the value of sin 240 degrees in fraction form?
−32-\frac{\sqrt{3}}{2}.
Why is sin 240 degrees negative?
Because 240°240° lies in Quadrant III, where the yy-coordinate on the unit circle is negative, and sine reads that yy-coordinate.
What is sin 240 degrees in radians?
sin⁡(4π3)=−32\sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}, since 240°=4π3240° = \frac{4\pi}{3}.
What is the reference angle for 240 degrees?
60°60°. For a Quadrant III angle the reference angle is the angle minus 180°180°.
Is sin 240 the same as sin 60?
Only in magnitude. sin⁡240°=−32\sin 240° = -\frac{\sqrt{3}}{2} and sin⁡60°=+32\sin 60° = +\frac{\sqrt{3}}{2}: same size, opposite sign.
✍️ Written By
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Bhanzu Team
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Bhanzu’s editorial team, known as Team Bhanzu, is made up of experienced educators, curriculum experts, content strategists, and fact-checkers dedicated to making math simple and engaging for learners worldwide. Every article and resource is carefully researched, thoughtfully structured, and rigorously reviewed to ensure accuracy, clarity, and real-world relevance. We understand that building strong math foundations can raise questions for students and parents alike. That’s why Team Bhanzu focuses on delivering practical insights, concept-driven explanations, and trustworthy guidance-empowering learners to develop confidence, speed, and a lifelong love for mathematics.
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