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.
Angle (radians) | Angle (degrees) | Quadrant | $\sin\theta$ (exact) | $\sin\theta$ (decimal) |
|---|---|---|---|---|
$\pi$ | $180^\circ$ | Axis | $0$ | $0.0000$ |
$\dfrac{7\pi}{6}$ | $210^\circ$ | Third | $-\dfrac{1}{2}$ | $-0.5000$ |
$\dfrac{5\pi}{4}$ | $225^\circ$ | Third | $-\dfrac{\sqrt{2}}{2}$ | $-0.7071$ |
$\dfrac{4\pi}{3}$ | $240^\circ$ | Third | $-\dfrac{\sqrt{3}}{2}$ | $-0.8660$ |
$\dfrac{3\pi}{2}$ | $270^\circ$ | Axis | $-1$ | $-1.0000$ |
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
Evaluate $2\sin\frac{4\pi}{3} + 1$.
State the reference angle and the sign of sine for $\frac{5\pi}{3}$, then find $\sin\frac{5\pi}{3}$.
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.
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