Sin 120 Degrees : Exact Value √3/2 and How to Find It

What Does Sin 120 Degrees Mean?

Sine is one of the three core trigonometric ratios. For an acute angle in a right triangle, the sine is the side opposite the angle divided by the hypotenuse. But $120^\circ$ is obtuse, so a right triangle can't hold it directly. That is where the unit circle takes over.

On the unit circle (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. Rotate $120^\circ$ counterclockwise from the positive x-axis and the radius lands at $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. The y-coordinate is $\frac{\sqrt{3}}{2}$, so that is $\sin 120^\circ$. The x-coordinate is negative and the y-coordinate is positive, which is exactly what "second quadrant" means for sin, cos, and tan.

Where Does Sin 120 Degrees Show Up?

Three-phase electric power spaces its three voltage waveforms exactly $120^\circ$ apart, so the sine of $120^\circ$ is baked into how the grid delivers power to your home. The interior angle of a regular hexagon is also $120^\circ$, which is why honeycomb cells and hex nuts tile without gaps. In any obtuse-triangle problem solved with the applications of trigonometry, a $120^\circ$ angle appears the moment two sides open wider than a right angle.

Standard-Angle Reference Table

A quadrant is one of the four quarters the x- and y-axes cut the plane into, numbered counterclockwise from the top-right. The angle $120^\circ$ opens past $90^\circ$, so it lands in the second quadrant, where the sine (the y-coordinate) is still positive. Here are the standard angles from $0^\circ$ through $180^\circ$ in both degrees and radians.

Read down the sine column and it rises to $1$ at $90^\circ$, then mirrors back down. That mirror is the key fact: $120^\circ$ and $60^\circ$ carry the same sine, and $150^\circ$ matches $30^\circ$. The full picture lives in the trigonometric table of standard values.

How Do You Find The Exact Value Of Sin 120 Degrees?

The cleanest route uses a reference angle: the acute angle between the terminal side and the x-axis. For any angle it tells you the size of the sine, and the quadrant tells you the sign.

Method 1: The reference angle.

For an angle between $90^\circ$ and $180^\circ$, the reference angle is $180^\circ - \theta$.

$$180^\circ - 120^\circ = 60^\circ$$

In the second quadrant sine is positive, so the sign stays plus:

$$\sin 120^\circ = +\sin 60^\circ = \frac{\sqrt{3}}{2}$$

Method 2: The supplementary-angle identity.

The identity $\sin(180^\circ - \theta) = \sin\theta$ says an angle and its supplement share a sine. Writing $120^\circ$ as $180^\circ - 60^\circ$:

$$\sin 120^\circ = \sin(180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2}$$

This is why $\sin 120^\circ$ matches $\sin 60^\circ$ rather than $\sin 240^\circ$. The angle $240^\circ$ shares the same $60^\circ$ reference but sits in the third quadrant, so its sine is negative, $-\frac{\sqrt{3}}{2}$. You can compare the two directly on the sin 240 degrees page.

Method 3: The decimal check.

A calculator in degree mode returns $\sin(120) = 0.8660254\ldots$. Squaring $\frac{\sqrt{3}}{2}$ gives $\frac{3}{4}$, whose square root is that same $0.8660$, confirming the exact form.

Examples Of Sin 120 Degrees

Example 1

Evaluate $6\sin 120^\circ$.

$$6\sin 120^\circ = 6 \times \frac{\sqrt{3}}{2} = 3\sqrt{3} \approx 5.196$$

Example 2

Find $\sin 120^\circ$ using its reference angle.

Wrong attempt. A student takes the reference angle as $120^\circ - 90^\circ = 30^\circ$ and writes $\sin 120^\circ = \frac{1}{2}$.

Check it against the table: $\sin 120^\circ$ should sit near $0.87$, not $0.5$, because $120^\circ$ is close to $90^\circ$ where sine peaks. So $\frac{1}{2}$ is too small to be right.

Correct. The reference angle is measured from the x-axis, not the y-axis, so it is $180^\circ - 120^\circ = 60^\circ$. Then $\sin 120^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866$. The mismatch students hit most is measuring the reference angle from the wrong axis.

Example 3

A triangle has an obtuse angle of $120^\circ$. Its longest side is $14$ cm, opposite the $120^\circ$ angle, and by the law of sines $\frac{\sin 120^\circ}{14} = \frac{\sin B}{b}$. Compute $\sin 120^\circ$ as a decimal for the calculation.

$$\sin 120^\circ = \frac{\sqrt{3}}{2} \approx 0.8660$$

That decimal is what you plug into the law of sines; keeping the exact $\frac{\sqrt{3}}{2}$ is fine too, and more precise.

Example 4

Verify that $\sin 120^\circ = \sin 60^\circ$ using the supplementary identity.

$$\sin 120^\circ = \sin(180^\circ - 60^\circ) = \sin 60^\circ$$

Both equal $\frac{\sqrt{3}}{2}$, so the identity holds.

Example 5

Express $\sin 120^\circ$ in radians and evaluate $\sin\left(\frac{2\pi}{3}\right)$.

Since $120^\circ = \frac{2\pi}{3}$ radians, $\sin\left(\frac{2\pi}{3}\right) = \sin 120^\circ = \frac{\sqrt{3}}{2}$. The degree form and the radian form name one angle and one value.

Where Students Trip Up On Sin 120 Degrees

Mistake 1: Making the sine negative

Where it slips in: Recall that "second quadrant" flips the sign of something often gets over-applied to sine.

Don't do this: Writing $\sin 120^\circ = -\frac{\sqrt{3}}{2}$.

The correct way: In the second quadrant the y-coordinate is positive, so sine is positive: $\sin 120^\circ = +\frac{\sqrt{3}}{2}$. Only cosine and tangent turn negative in the second quadrant. The first-instinct error here is assuming every second-quadrant ratio is negative.

Mistake 2: Measuring the reference angle from the wrong axis

Where it slips in: Building the reference angle by subtracting $90^\circ$ instead of subtracting from $180^\circ$.

Don't do this: Using $120^\circ - 90^\circ = 30^\circ$ and reporting $\frac{1}{2}$.

The correct way: For a second-quadrant angle, reference angle $= 180^\circ - \theta = 60^\circ$. The reference angle is always the acute gap to the horizontal axis.

Mistake 3: Confusing sin 120° with cos 120°

Where it slips in: Reading a value off the point $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ and grabbing the wrong coordinate.

Don't do this: Writing $\sin 120^\circ = -\frac{1}{2}$, which is actually the x-coordinate, so it is $\cos 120^\circ$.

The correct way: Sine is the y-coordinate, $\frac{\sqrt{3}}{2}$; cosine is the x-coordinate, $-\frac{1}{2}$. When a $120^\circ$ answer comes out negative, the usual cause is that cosine got read where sine was wanted.

Key Takeaways

• Sin 120 degrees equals $\frac{\sqrt{3}}{2}$, about $0.8660$, an exact value because $120^\circ$ builds on the $60^\circ$ reference angle.

• The angle sits in the second quadrant, where sine stays positive; only its reference angle ($60^\circ$) sets the size.

• In radians, $\sin 120^\circ = \sin\left(\frac{2\pi}{3}\right)$, and by the supplementary identity $\sin 120^\circ = \sin 60^\circ$.

• The common slips are a stray negative sign and a reference angle measured from the wrong axis.

• To build this with a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or structured math classes online.

Practice These Before Moving On

1. Evaluate $\sin 120^\circ + \sin 150^\circ$ using exact values.

2. A parallelogram has a $120^\circ$ angle and sides $8$ and $5$. Its area is $8 \times 5 \times \sin 120^\circ$; find it exactly.

3. Show that $\sin 120^\circ = \cos 30^\circ$ and explain why using cofunctions.

Want a live Bhanzu trainer to walk through more sin 120 degrees problems? Book a free demo class.

Read More

• Sin 150 degrees — the other second-quadrant standard angle, equal to $\frac{1}{2}$.

• Sin 180 degrees — where the sine curve returns to zero.

• Sin (A − B) formula — build values like $\sin 120^\circ$ from angle differences.

• Pythagorean identities — the $\sin^2\theta + \cos^2\theta = 1$ relationship behind every unit-circle point.

• Cofunction identities — why $\sin 120^\circ = \cos 30^\circ$.