Sin 2π/3 : Exact Value √3/2 and How to Find It

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.

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

1. Evaluate $6\sin\frac{2\pi}{3} - 2$.

2. Find $\sin\frac{2\pi}{3} + \sin\frac{\pi}{3}$ and explain the result using reference angles.

3. 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.