Sin 4pi/3 : Exact Value -√3/2 and How to Find It

What Does Sin 4pi/3 Mean?

The angle $\frac{4\pi}{3}$ is a rotation of two-thirds of $\pi$ past the half-turn, landing in the third quadrant, the region past $180^\circ$ but before $270^\circ$. On the unit circle, sine is the $y$-coordinate of the point where the rotated radius meets the circle.

For $\frac{4\pi}{3}$ that point is $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$, so the sine is the $y$-value $-\frac{\sqrt{3}}{2}$. The size $\frac{\sqrt{3}}{2}$ is inherited from the sin 60° computation; only the sign changes.

Where Does Sin 4pi/3 Show Up?

Angles beyond $\pi$ appear whenever motion keeps rotating past half a turn: a Ferris-wheel car at $\frac{4\pi}{3}$ of its cycle sits low and to one side, its height below the axle proportional to $\sin\frac{4\pi}{3}$. Alternating current and any rotating phasor pass through $\frac{4\pi}{3}$ once per cycle, and there the signal is at $-\frac{\sqrt{3}}{2}$ of its peak.

The negative value is the whole point: it tells you the direction, not just the size, of the vertical component. That sign is read directly from the unit circle, where third-quadrant points fall below the horizontal axis.

Standard-Angle Reference Table

An angle written in radians measures arc length on a unit circle rather than degrees. Here is $\frac{4\pi}{3}$ among the third-quadrant standard angles, with the corresponding sine values.

Every sine value in the third quadrant is negative, because the $y$-coordinate sits below the $x$-axis. The degree twin of this angle, sin 240°, leads with the degree computation of the same value.

How Do You Find The Exact Value Of Sin 4pi/3?

The cleanest route uses the reference angle, then applies the quadrant sign separately.

Method 1: Reference angle plus quadrant sign.

The reference angle is the acute angle between the terminal radius and the $x$-axis. For a third-quadrant angle you subtract $\pi$:

$$\frac{4\pi}{3} - \pi = \frac{\pi}{3}$$

The sine of the reference angle is $\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$. Sine is negative in the third quadrant, so attach the minus sign:

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

Method 2: The unit circle point.

Rotate the radius $\frac{4\pi}{3}$ (that is, $240^\circ$) from the positive $x$-axis. The tip lands at $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$, and the sine is the $y$-coordinate:

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

Both methods return $-\frac{\sqrt{3}}{2} \approx -0.8660$, because the reference-angle rule is just a shortcut for reading the same unit-circle point.

Examples Of Sin 4pi/3

Example 1

Evaluate $4\sin\frac{4\pi}{3}$.

$$4\sin\frac{4\pi}{3} = 4 \times \left(-\frac{\sqrt{3}}{2}\right) = -2\sqrt{3} \approx -3.464$$

Example 2

A student says $\sin\frac{4\pi}{3} = +\frac{\sqrt{3}}{2}$ because the reference angle is $\frac{\pi}{3}$.

Wrong attempt. The student finds the reference angle correctly, then copies the first-quadrant sign along with it.

That skips a step: the reference angle fixes the size, but the quadrant fixes the sign. A third-quadrant $y$-coordinate is below the axis, so a positive answer contradicts the picture.

Correct. Take the size from the reference angle, $\frac{\sqrt{3}}{2}$, then apply the third-quadrant rule that sine is negative: $\sin\frac{4\pi}{3} = -\frac{\sqrt{3}}{2}$.

Example 3

Verify $\sin^2\frac{4\pi}{3} + \cos^2\frac{4\pi}{3} = 1$, given $\cos\frac{4\pi}{3} = -\frac{1}{2}$.

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

The Pythagorean identity holds regardless of the signs, since squaring removes them.

Example 4

Express $\sin\frac{4\pi}{3}$ using the periodicity $\sin(\theta + 2\pi) = \sin\theta$, then relate it to $\sin\frac{\pi}{3}$.

Adding $2\pi$ gives $\frac{4\pi}{3} + 2\pi = \frac{10\pi}{3}$, so $\sin\frac{10\pi}{3} = \sin\frac{4\pi}{3} = -\frac{\sqrt{3}}{2}$. The angle $\frac{4\pi}{3}$ is the third-quadrant reflection of $\frac{\pi}{3}$, which is why the magnitude matches while the sign flips.

Example 5

Find $\sin\frac{4\pi}{3} - \sin\frac{5\pi}{4}$.

$$\sin\frac{4\pi}{3} - \sin\frac{5\pi}{4} = -\frac{\sqrt{3}}{2} - \left(-\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2} - \sqrt{3}}{2} \approx -0.159$$

The neighbouring third-quadrant value comes from sin 5π/4, whose reference angle is $\frac{\pi}{4}$.

Where Students Trip Up On Sin 4pi/3

Mistake 1: Keeping the positive sign of the reference angle

Where it slips in: After finding the reference angle $\frac{\pi}{3}$, the sign of the original quadrant gets forgotten.

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

The correct way: Read the size from the reference angle, then set the sign from the quadrant. Third quadrant means sine is negative, so $-\frac{\sqrt{3}}{2}$. The learner who treats the reference angle as the whole answer is the one who loses the sign every time.

Mistake 2: Subtracting from the wrong benchmark

Where it slips in: Computing the reference angle by subtracting $\frac{\pi}{2}$ or $2\pi$ instead of $\pi$.

Don't do this: Writing the reference angle as $\frac{4\pi}{3} - \frac{\pi}{2} = \frac{5\pi}{6}$, which is not acute.

The correct way: In the third quadrant the reference angle is $\theta - \pi$, so $\frac{4\pi}{3} - \pi = \frac{\pi}{3}$. A reference angle must be acute; if the result is larger than $\frac{\pi}{2}$, the wrong benchmark was used.

Mistake 3: Reading the calculator in degree mode

Where it slips in: Entering $\sin(4\pi/3)$ with the calculator set to degrees returns roughly $0.073$, not $-0.866$.

Don't do this: Trusting the screen without matching the mode to the angle's unit.

The correct way: A radian angle needs radian mode. When the answer is close to $0$ for an angle that should be near a peak, the mode is usually wrong.

Key Takeaways

• Sin 4pi/3 equals $-\frac{\sqrt{3}}{2}$, about $-0.8660$, an exact value because $\frac{4\pi}{3}$ is a standard angle.

• The reference angle is $\frac{\pi}{3}$, giving the size $\frac{\sqrt{3}}{2}$; the third-quadrant rule supplies the negative sign.

• In degrees, $\frac{4\pi}{3} = 240^\circ$, and the unit-circle point is $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$.

• The most common slip is keeping the positive reference-angle value; sine is negative below the $x$-axis.

To build quadrant reasoning with a teacher, explore Bhanzu's trigonometry tutor or high school math tutor, or start with structured math tutoring.

Practice These Before Moving On

1. Evaluate $2\sin\frac{4\pi}{3} + 1$.

2. State the reference angle and the sign of sine for $\frac{5\pi}{3}$, then find $\sin\frac{5\pi}{3}$.

3. Show that $\sin\frac{4\pi}{3} = \sin\left(\pi + \frac{\pi}{3}\right)$ using the identity $\sin(\pi + x) = -\sin x$.

Want a live Bhanzu trainer to walk through more sin 4pi/3 problems? Book a free demo class.

Read More

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