Sin Pi/3 : Exact Value √3/2 on the Unit Circle

What Does Sin Pi/3 Mean?

Sine is one of the three trigonometric ratios: in a right triangle it is the side opposite the angle divided by the hypotenuse. So $\sin\frac{\pi}{3}$ asks what fraction of the hypotenuse the opposite side reaches when the angle is $60^\circ$.

On the unit circle, sine is the $y$-coordinate of the point where the angle's radius meets the circle. Sweeping through $\frac{\pi}{3}$ radians (a radian being the arc-equals-radius angle) lands the radius at $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, so the height, and therefore sin pi/3, is $\frac{\sqrt{3}}{2}$.

Where Does Sin Pi/3 Show Up?

A $\frac{\pi}{3}$ angle (that is, $60^\circ$) is the interior angle of an equilateral triangle, so sin pi/3 governs the height of anything built from those triangles. The height of an equilateral triangle with side $s$ is $s\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}s$, which sets the geometry of a hexagonal tile floor and the trusses in a triangular bridge frame.

The value also appears in three-phase electricity, where the phases sit $\frac{\pi}{3}$ radians apart, and in vector problems where a force at $60^\circ$ contributes $\frac{\sqrt{3}}{2}$ of its size upward. Wherever a $60^\circ$ tilt sends most of a quantity in one direction, $\frac{\sqrt{3}}{2}$ is the fraction that survives.

What Is The Standard-Angle Sine Reference Table?

The angle $\frac{\pi}{3}$ radians is a third of a half-turn, and it sits high on the sine climb. Reading the table in radian order, $\sin\theta$ rises from $0$ toward $1$, and $\frac{\pi}{3}$ is the near-top value $\frac{\sqrt{3}}{2}$.

Notice the mirror: sin pi/3 equals the cosine of its complement, since $\sin\frac{\pi}{3} = \cos\frac{\pi}{6}$. That cofunction swap is why $\frac{\sqrt{3}}{2}$ shows up for both $60^\circ$ sine and $30^\circ$ cosine.

How Do You Find The Exact Value Of Sin Pi/3?

Two routes reach $\frac{\sqrt{3}}{2}$: build it from a triangle, or read it off the circle.

Method 1: The 30-60-90 triangle.

Take an equilateral triangle with side $2$ and drop a perpendicular from one vertex to the opposite side, splitting it into two right triangles with angles $\angle 30^\circ$, $\angle 60^\circ$, and $\angle 90^\circ$.

In one of them the hypotenuse is $2$, the side opposite the $60^\circ$ angle is $\sqrt{3}$ (from $\sqrt{2^2 - 1^2}$), and the side opposite $30^\circ$ is $1$.

$$\sin\frac{\pi}{3} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}$$

Method 2: The unit circle.

Rotate a radius of length $1$ through $\frac{\pi}{3}$ radians.

Its tip lands at $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

$$\sin\frac{\pi}{3} = y\text{-coordinate} = \frac{\sqrt{3}}{2}$$

Method 3: The calculator check.

In radian mode, $\sin(\pi \div 3)$ returns $0.8660254\ldots$. Squaring $\frac{\sqrt{3}}{2}$ gives $\frac{3}{4}$, whose square root is that same $0.8660$, confirming the exact form.

Examples Of Sin Pi/3

Example 1

Evaluate $2\sin\left(\frac{\pi}{3}\right)$.

$$2\sin\left(\frac{\pi}{3}\right) = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3} \approx 1.732$$

Example 2

Evaluate $\sin\left(\frac{\pi}{3}\right)$ and state which is larger, $\sin\frac{\pi}{3}$ or $\sin\frac{\pi}{6}$.

Wrong attempt. A student writes $\sin\frac{\pi}{3} = \frac{1}{2}$, thinking the "$3$" in the denominator pairs with the smaller value.

That reverses the pair. The larger acute angle has the larger sine, and $\frac{\pi}{3}$ (which is $60^\circ$) is larger than $\frac{\pi}{6}$ (which is $30^\circ$), so its sine must be bigger, not smaller.

Correct. $\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} \approx 0.87$, while $\sin\frac{\pi}{6} = \frac{1}{2} = 0.5$, so sin pi/3 is the larger. Compare the smaller partner in sin pi/6.

Example 3

Verify the identity $\sin^2\left(\frac{\pi}{3}\right) + \cos^2\left(\frac{\pi}{3}\right) = 1$.

Since $\cos\frac{\pi}{3} = \frac{1}{2}$:

$$\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = \frac{3}{4} + \frac{1}{4} = 1$$

The Pythagorean identity holds.

Example 4

A right triangle has a hypotenuse of $8$ cm and a $\frac{\pi}{3}$ angle. Find the side opposite that angle.

$$\sin\frac{\pi}{3} = \frac{\text{opposite}}{8} \implies \text{opposite} = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3} \approx 6.93 \text{ cm}$$

Example 5

Evaluate $\sin\left(\frac{4\pi}{3}\right)$ using the reference angle.

The angle $\frac{4\pi}{3}$ sits in the third quadrant, where sine is negative, and its reference angle back to the horizontal axis is $\frac{\pi}{3}$.

$$\sin\frac{4\pi}{3} = -\sin\frac{\pi}{3} = -\frac{\sqrt{3}}{2}$$

The degree form of this same third-quadrant value is worked in sin 240 degrees.

Where Students Trip Up On Sin Pi/3

Mistake 1: Swapping sin pi/3 and sin pi/6

Where it slips in: Recall under time pressure, when the $\frac{\sqrt{3}}{2}$ and the $\frac{1}{2}$ get attached to the wrong angle.

Don't do this: Writing $\sin\frac{\pi}{3} = \frac{1}{2}$, which is actually $\sin\frac{\pi}{6}$.

The correct way: Bigger acute angle, bigger sine. $\frac{\pi}{3}$ is $60^\circ$, so it takes the larger value $\frac{\sqrt{3}}{2}$; $\frac{\pi}{6}$ is $30^\circ$ and takes $\frac{1}{2}$. Students first meeting the special angles tend to pair the numbers by the denominator instead of by size, so anchor on the angle, not the fraction.

Mistake 2: Forgetting the quadrant sign

Where it slips in: Extending sin pi/3 to related angles like $\frac{4\pi}{3}$ or $\frac{2\pi}{3}$ without checking the quadrant.

Don't do this: Writing $\sin\frac{4\pi}{3} = \frac{\sqrt{3}}{2}$.

The correct way: The reference angle gives the size $\frac{\sqrt{3}}{2}$; the quadrant supplies the sign separately. In the third quadrant sine is negative, so $\sin\frac{4\pi}{3} = -\frac{\sqrt{3}}{2}$.

Mistake 3: Rounding when an exact value is asked

Where it slips in: Calculator-first solving, where the screen reads $0.866$ and the student copies it.

Don't do this: Writing $\sin\frac{\pi}{3} = 0.866$ on a problem that asks for the exact value.

The correct way: Give the radical form $\frac{\sqrt{3}}{2}$. The decimal $0.8660$ is a rounded approximation; $\frac{\sqrt{3}}{2}$ is the exact value.

Key Takeaways

• Sin pi/3 equals $\frac{\sqrt{3}}{2}$, about $0.8660$, an exact value because $\frac{\pi}{3}$ is a standard angle.

• The 30-60-90 triangle gives it as opposite over hypotenuse; the unit circle gives it as the $y$-coordinate at $\frac{\pi}{3}$.

• In degrees, $\sin\frac{\pi}{3} = \sin 60^\circ = \frac{\sqrt{3}}{2}$.

• The most common slip is swapping it with $\sin\frac{\pi}{6} = \frac{1}{2}$: the bigger angle takes the bigger value.

To master the standard angles 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 $4\sin\left(\frac{\pi}{3}\right) - \cos\left(\frac{\pi}{3}\right)$.

2. An equilateral triangle has side $6$ cm. Use $\sin\frac{\pi}{3}$ to find its height.

3. Evaluate $\sin\left(\frac{2\pi}{3}\right)$ using the reference angle, and state its sign.

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Read More

• Sin 60 degrees — the degree twin of this angle, same value $\frac{\sqrt{3}}{2}$.

• The full trigonometric table — every standard angle in one place.

• 60 degrees to radians — the conversion behind $\frac{\pi}{3} = 60^\circ$.

• Cofunction identities — why $\sin\frac{\pi}{3} = \cos\frac{\pi}{6}$.

• Applications of trigonometry — where these values do real work.