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

TL;DR

The value of sin 240 degrees is exactly −√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.

The value of sin 240 degrees is −√3/2 ≈ −0.8660.

Quick Answer:

Result: sin⁡240°=−√3/2

Decimal: −0.8660 (to four places)

In radians: sin⁡(4π/3)=−√3/2

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

Method shown: reference angle, Quadrant III sign

Quick Reference Table of Sine Values

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 (−1/2,−√3/2) whose negative y-coordinate gives sin⁡240°=−√3/2.

How to Find the Value of Sin 240 Degrees

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°=√3/2. Now fix the sign. The angle 240° is in Quadrant III, where sine is negative:

sin⁡240°=−sin⁡60°=−√3/2.

Method 2: Unit circle

Rotate the unit radius 240° counterclockwise. It lands at (−1/2,−√3/2), and the sine is the y-coordinate:

sin⁡240°=y-coordinate=−√3/2.

Method 3: Supplementary-style identity

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

sin⁡240°=sin⁡(180°+60°)=−sin⁡60°=−√3/2.

Examples of Sin 240 Degrees

Example 1: Evaluate 2sin⁡240°

2sin⁡240°=2⋅(−√3/2)=−√3≈−1.732.

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

The quick instinct is to subtract from 360°: 360°−240°=120°. That is wrong, because 120° is not acute. For a Quadrant III angle you subtract 180°:

240°−180°=60°.

Thus, sin⁡240°=−sin⁡60°=−√3/2.

Example 3: Evaluate sin⁡240°+sin⁡60°

−√3/2+√3/2=0.

Example 4: Verify sin²240°+cos²240°=1

(−√3/2)²+(−1/2)²=1.

Example 5

Express 240° in radians and state the value. 240°=4π/3 radians, so sin⁡(4π/3)=−√3/2.

Common Mistakes With Sin 240 Degrees

Mistake 1: Dropping the negative sign

Correct way: sin⁡240°=−√3/2.

Mistake 2: Using the wrong reference-angle rule

Correct way: Quadrant III uses θ−180°, giving 60°.

Mistake 3: Confusing the reference angles of 240° and 210°

Correct way: They land in the same quadrant with different reference angles.

Key Takeaways

• Sin 240 degrees equals −√3/2 (about −0.8660), an exact value because 240° is a standard angle.
• The reference angle is 60°, giving the magnitude √3/2; Quadrant III makes it negative.
• In radians, sin⁡240°=sin⁡(4π/3).
• The most common error is dropping the negative sign — always read the quadrant before the magnitude.