What Does Sin 2π/3 Mean?
An angle in radians is the arc length on a unit circle, and $\frac{2\pi}{3}$ is two-thirds of the way from $0$ to a half-turn, which is $120^\circ$. Before finding its sine, name the quadrant: the four quarters of the plane, numbered anticlockwise from the top-right. Since $\frac{\pi}{2} < \frac{2\pi}{3} < \pi$, the angle lands in Quadrant II, the top-left, where the $y$-coordinate is positive and so sine is positive.
The reference angle is the acute angle between the terminal radius and the $x$-axis. In Quadrant II it is $\pi - \frac{2\pi}{3} = \frac{\pi}{3}$. The reference angle carries the size of the value; the quadrant carries the sign. So:
$$\sin\frac{2\pi}{3} = +\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$$
On the unit circle the terminal point is $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$; sine reads the $y$-coordinate $\frac{\sqrt{3}}{2}$, while cos 2π/3 reads the negative $x$-coordinate $-\frac{1}{2}$.
Where Does Sin 2π/3 Show Up?
A $120^\circ$ angle is the natural spacing of three things arranged evenly around a centre, and $\sin\frac{2\pi}{3}$ measures the height of the arms. The three phases of a mains electricity supply are set $120^\circ$ apart, so each phase's voltage at a given instant involves $\sin\frac{2\pi}{3}$. The same $120^\circ$ appears between the bonds of a flat carbon ring, between the spokes of a three-armed rotor, and between the legs of a stable tripod. Whenever something splits a full turn into three equal parts, the $\frac{\sqrt{3}}{2}$ value is close by.
What Is The Value Of Sin 2π/3?
Sin 2π/3 is exactly $\frac{\sqrt{3}}{2}$, the same height as $\sin\frac{\pi}{3}$, just reached in the second quadrant instead of the first. Here are the standard angles around it in both units, so you can see where $\frac{2\pi}{3}$ sits.
Angle (radians) | Angle (degrees) | $\sin\theta$ (exact) | $\sin\theta$ (decimal) |
|---|---|---|---|
$\dfrac{\pi}{3}$ | $60^\circ$ | $\dfrac{\sqrt{3}}{2}$ | $0.8660$ |
$\dfrac{\pi}{2}$ | $90^\circ$ | $1$ | $1.0000$ |
$\dfrac{2\pi}{3}$ | $120^\circ$ | $\dfrac{\sqrt{3}}{2}$ | $0.8660$ |
$\dfrac{5\pi}{6}$ | $150^\circ$ | $\dfrac{1}{2}$ | $0.5000$ |
$\pi$ | $180^\circ$ | $0$ | $0.0000$ |
Sine rises to $1$ at $90^\circ$, then falls back. At $120^\circ$ it has dropped to $\frac{\sqrt{3}}{2}$ again, mirroring its value at $60^\circ$. That mirror is exactly the reference-angle relationship: $\frac{2\pi}{3}$ and $\frac{\pi}{3}$ share a sine because they are the same distance from the horizontal axis. This is why sin 2π/3 and sin π/3 print the identical value.
How Do You Find The Exact Value Of Sin 2π/3?
Two clean routes, both giving $\frac{\sqrt{3}}{2}$.
Method 1: Reference angle and quadrant sign.
Convert to degrees if it helps: $\frac{2\pi}{3} = \frac{2\pi}{3} \times \frac{180^\circ}{\pi} = 120^\circ$. The angle is in Quadrant II, so sine is positive, and the reference angle is:
$$180^\circ - 120^\circ = 60^\circ \quad\Rightarrow\quad \sin 120^\circ = +\sin 60^\circ = \frac{\sqrt{3}}{2}$$
The degree form $\sin 60^\circ$ is the same value as sin 60 degrees from the 30-60-90 triangle.
Method 2: The supplementary-angle identity.
Sine of an angle equals sine of its supplement: $\sin\theta = \sin(\pi - \theta)$. With $\theta = \frac{2\pi}{3}$:
$$\sin\frac{2\pi}{3} = \sin\left(\pi - \frac{2\pi}{3}\right) = \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$$
Same answer, reached by algebra rather than a picture.
Examples Of Sin 2π/3
Example 1
Evaluate $4\sin\dfrac{2\pi}{3}$.
$$4\sin\frac{2\pi}{3} = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3} \approx 3.464$$
Example 2
Find $\sin\dfrac{2\pi}{3}$.
Wrong attempt. A student sees the angle is past $\frac{\pi}{2}$, decides it is "in the negative part," and writes $\sin\frac{2\pi}{3} = -\frac{\sqrt{3}}{2}$.
Check the picture. Quadrant II is the top-left, above the $x$-axis, so heights there are positive. A negative sine would put the point below the axis, which is Quadrant III or IV.
Correct. In Quadrant II sine is positive. Apply the reference angle $\frac{\pi}{3}$ with a positive sign:
$$\sin\frac{2\pi}{3} = +\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$$
Example 3
Verify $\sin^2\dfrac{2\pi}{3} + \cos^2\dfrac{2\pi}{3} = 1$.
$$\left(\frac{\sqrt{3}}{2}\right)^2 + \left(-\frac{1}{2}\right)^2 = \frac{3}{4} + \frac{1}{4} = 1$$
The Pythagorean identity holds even though cosine is negative here.
Example 4
A triangle has an angle of $\dfrac{2\pi}{3}$ opposite a side of length $a$, with circumradius $R$. Using the law of sines relation $a = 2R\sin A$, find $a$ when $R = 5$.
$$a = 2 \times 5 \times \sin\frac{2\pi}{3} = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66$$
Example 5
Express sin 2π/3 in degrees and evaluate.
Since $\frac{2\pi}{3} = 120^\circ$,
$$\sin 120^\circ = \sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2}$$
The radian and degree forms name one angle and one value.
Where Students Trip Up On Sin 2π/3
Mistake 1: Making The Sine Negative In Quadrant II
Where it slips in: Seeing the angle exceed $\frac{\pi}{2}$ and assuming the value must turn negative.
Don't do this: Writing $\sin\frac{2\pi}{3} = -\frac{\sqrt{3}}{2}$.
The correct way: Sine is positive in Quadrants I and II. The first-instinct error is to attach a negative sign the moment an angle passes $90^\circ$; check the quadrant instead, and above the $x$-axis sine stays positive.
Mistake 2: Using The Wrong Reference Angle
Where it slips in: Computing the reference angle as $\frac{2\pi}{3} - \frac{\pi}{2}$ or $\frac{2\pi}{3}$ itself.
Don't do this: Taking the reference angle as $\frac{\pi}{6}$ (which would give $\frac{1}{2}$).
The correct way: In Quadrant II the reference angle is $\pi$ minus the angle, so $\pi - \frac{2\pi}{3} = \frac{\pi}{3}$. Skipping this step is exactly where the value comes out wrong, even when the sign is right.
Mistake 3: Halving The Angle Into 2π/3 Carelessly
Where it slips in: Reading $\frac{2\pi}{3}$ as "a third of $2\pi$" and confusing it with a full-turn fraction.
Don't do this: Treating $\frac{2\pi}{3}$ as coterminal with $\frac{\pi}{3}$ by dropping the $2$.
The correct way: $\frac{2\pi}{3}$ is $120^\circ$, not $60^\circ$; the two share a sine only through the reference-angle rule, not because the angles are equal. The habit of converting to degrees first keeps the two apart.
Key Takeaways
Sin 2π/3 equals $\frac{\sqrt{3}}{2}$, about $0.8660$, an exact value from the reference angle $\frac{\pi}{3}$.
The angle is $120^\circ$ in Quadrant II, where sine is positive, so no negative sign appears.
It shares its value with $\sin\frac{\pi}{3}$ and $\sin 60^\circ$; the terminal point is $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.
The most common slips are making the sine negative or taking the reference angle as $\frac{\pi}{6}$.
To master quadrants and reference angles with a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or math tutoring online.
Practice These Before Moving On
Evaluate $6\sin\frac{2\pi}{3} - 2$.
Find $\sin\frac{2\pi}{3} + \sin\frac{\pi}{3}$ and explain the result using reference angles.
Using the core trigonometric ratios, find $\tan\frac{2\pi}{3}$ from $\sin\frac{2\pi}{3}$ and $\cos\frac{2\pi}{3}$.
Want a live Bhanzu trainer to walk through more sin 2π/3 problems? Book a free demo class.
Read More
Sin, cos, and tan explained: the ratios behind every angle value.
Trigonometric table: standard-angle sines and cosines at a glance.
Sin 2π: the full-turn value, for contrast with this second-quadrant one.
Cofunction identities: how supplementary and complementary angles share values.
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