What Is The Value Of Sin 330 Degrees?
Sin 330 degrees is $-\dfrac{1}{2}$, or $-0.5$ as a decimal. Written in radians, the angle 330° is $\dfrac{11\pi}{6}$, so $\sin 330^\circ = \sin\dfrac{11\pi}{6} = -\dfrac{1}{2}$.
This is an exact value, not a rounded approximation. The fraction $-\frac{1}{2}$ is complete on its own, and the decimal $-0.5$ happens to terminate, so both forms are precise. Two facts do all the work: 330° sits in the fourth quadrant, where sine is negative, and it is exactly 30° short of a full turn, so its size matches $\sin 30^\circ = \frac{1}{2}$.
$$\sin 330^\circ = \sin\frac{11\pi}{6} = -\frac{1}{2} = -0.5$$
How Do You Find Sin 330 Degrees?
Every "sine of an awkward angle" question reduces to two steps: find the reference angle, then fix the sign from the quadrant. Sin 330 degrees is a clean example of both.
Step 1, locate the quadrant. 330° lies between 270° and 360°, so it is in the fourth quadrant. In that quadrant the sine ratio is negative.
Step 2, find the reference angle. The reference angle is the gap to the nearest horizontal axis. For a fourth-quadrant angle that is $360^\circ - 330^\circ = 30^\circ$.
Step 3, read the base value. The sine of the reference angle is $\sin 30^\circ = \frac{1}{2}$.
Step 4, apply the sign. Fourth quadrant means negative, so $\sin 330^\circ = -\sin 30^\circ = -\frac{1}{2}$.
A memory aid keeps the signs straight. The ASTC rule (read anticlockwise from the first quadrant: All, Sine, Tangent, Cosine) tells you which ratio stays positive in each quadrant. The fourth quadrant is the "C" quadrant, where only cosine (and its reciprocal) is positive, so sine there is negative.
For the full set of special-angle values behind this method, the trigonometric ratios of specific angles page lists them in one place, and the trigonometric table gives the standard grid.
Where Does 330 Degrees Sit On The Unit Circle?
On the unit circle, an angle is measured anticlockwise from the positive $x$-axis, and the point where the terminal side crosses the circle has coordinates $(\cos\theta, \sin\theta)$. The sine of the angle is simply the $y$-coordinate of that point.
At 330° the terminal side points down and to the right, landing in the fourth quadrant. Its coordinates are:
$$(\cos 330^\circ, \sin 330^\circ) = \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) \approx (0.8660, -0.5000)$$
The $x$-coordinate is positive (the point is to the right of centre) and the $y$-coordinate is negative (the point is below centre). That negative height is $\sin 330^\circ = -\frac{1}{2}$, read straight off the circle.
How Do You See Sin 330 Degrees In A Right Triangle?
The unit circle gives the sign, but the size, $\frac{1}{2}$, comes from an ordinary right triangle. Drop the vertical line from the 330° point to the $x$-axis and you form a right triangle whose acute angle at the origin is the 30° reference angle.
In a 30-60-90 triangle the side opposite the 30° angle is exactly half the hypotenuse. On the unit circle the hypotenuse is the radius, which is 1, so the opposite side has length $\frac{1}{2}$.
$$\sin 30^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1/2}{1} = \frac{1}{2}$$
The triangle only measures lengths, which are always positive, so it delivers the magnitude $\frac{1}{2}$. The unit circle then supplies the direction: the point is below the axis, so the signed value is $-\frac{1}{2}$. Size from the triangle, sign from the circle, and the two agree. If the definitions of these ratios need a refresher, the sin cos tan page walks through them from the triangle up.
How Can You Derive Sin 330 Degrees With The Angle-Difference Formula?
The reference-angle method is fast, but the sine subtraction formula proves the same result from first principles. Write 330° as $360^\circ - 30^\circ$ and expand:
$$\sin(360^\circ - 30^\circ) = \sin 360^\circ \cos 30^\circ - \cos 360^\circ \sin 30^\circ$$
Substitute the known values $\sin 360^\circ = 0$, $\cos 360^\circ = 1$, $\sin 30^\circ = \frac{1}{2}$, and $\cos 30^\circ = \frac{\sqrt{3}}{2}$:
$$= (0)\left(\frac{\sqrt{3}}{2}\right) - (1)\left(\frac{1}{2}\right)$$
$$= -\frac{1}{2}$$
A second route confirms it. Writing 330° as $270^\circ + 60^\circ$ and expanding the sine addition formula gives $\sin 270^\circ \cos 60^\circ + \cos 270^\circ \sin 60^\circ = (-1)\left(\frac{1}{2}\right) + (0)\left(\frac{\sqrt{3}}{2}\right) = -\frac{1}{2}$. Both routes land on $-\frac{1}{2}$, matching the unit circle exactly.
What Are The Related Values Around Sin 330 Degrees?
Four angles share the 30° reference angle, one in each quadrant. Comparing them side by side makes the sign pattern obvious, since only the quadrant changes the sign, never the digits.
Table: The 30°-reference family across all four quadrants, in degrees and radians.
Angle | Radians | sin | cos | tan |
|---|---|---|---|---|
30° | $\frac{\pi}{6}$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ |
150° | $\frac{5\pi}{6}$ | $\frac{1}{2}$ | $-\frac{\sqrt{3}}{2}$ | $-\frac{1}{\sqrt{3}}$ |
210° | $\frac{7\pi}{6}$ | $-\frac{1}{2}$ | $-\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ |
330° | $\frac{11\pi}{6}$ | $-\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $-\frac{1}{\sqrt{3}}$ |
The sine column is positive in the first two rows (quadrants I and II, above the axis) and negative in the last two (quadrants III and IV, below the axis). Sin 210 degrees and sin 330 degrees are equal at $-\frac{1}{2}$, one in the third quadrant and one in the fourth, which is worth remembering as a pair. For these values expressed in radian arguments, see trigonometric ratios in radians.
Why Is Sin 330 Degrees Negative?
The sign of sine is not a rule to memorise in isolation. It follows directly from where the angle points on the unit circle.
Sine is a height. The value $\sin\theta$ is the $y$-coordinate of the point on the unit circle. Above the $x$-axis the height is positive, below it the height is negative.
330° points downward. A 330° sweep ends in the fourth quadrant, below the horizontal axis, so its $y$-coordinate, and therefore its sine, is negative.
The size is set by the reference angle. The point is 30° short of a full turn, so its height matches the height at 30°, which is $\frac{1}{2}$. Combine the negative direction with the size $\frac{1}{2}$ and the value is $-\frac{1}{2}$.
The same logic explains every angle between 180° and 360°: they all dip below the axis, so their sines are all negative. Sin 330 degrees is simply one member of that lower half of the circle.
Who Discovered The Sine Function?
The sine we use today began not as a ratio of triangle sides but as a table of chord lengths, built by astronomers who needed to predict where planets and stars would appear.
Two other figures shaped the same idea:
Hipparchus of Nicaea (c. 190–120 BCE, Greece) built the first known table of chords, the direct ancestor of the sine table, to do astronomy.
Madhava of Sangamagrama (c. 1340–1425, India) found the infinite power series for sine, the method calculators still lean on to produce values like $-0.5$ for any angle.
Where Is Sin 330 Degrees Used In The Real World?
Any quantity that rises and falls in a smooth cycle is described by a sine, and the negative part of the cycle, where sin 330 degrees lives, is just as real as the positive part.
Alternating current. Household electricity is a sine wave, and for part of every cycle the voltage is negative, meaning the current flows the other way. A phase of 330° corresponds to a point in that below-zero stretch.
Tides and waves. The height of the sea above or below its average level follows a sine curve, and a negative value simply means the water is below the mean line, which is low tide territory.
Ferris wheels and circular motion. A rider's height relative to the wheel's centre is $R\sin\theta$. At 330° the rider is below the centre and descending, giving a negative height, exactly the scene in the opening image.
Sound and music. A pure tone is a sine wave in air pressure; the negative half of each cycle is the rarefaction, where pressure dips below normal.
Computer graphics. Rotating a point by 330° uses $\sin 330^\circ = -\frac{1}{2}$ inside the rotation formula, so the value is doing quiet arithmetic behind on-screen animation.
One value, $-\frac{1}{2}$, threads through power grids, oceans, fairground rides, and music, because all of them are cycles, and sine is the mathematics of a cycle.
What Are The Most Common Mistakes With Sin 330 Degrees?
These four errors account for most wrong answers on angles like 330°, confirmed against student answer threads on this exact angle and its neighbours.
Giving a positive answer.
Where it slips in:
A student finds the reference-angle value $\sin 30^\circ = \frac{1}{2}$ and writes it as the final answer, forgetting the sign.
Don't do this:
Do not stop at the reference angle. The reference angle only gives the size, never the sign.
The correct way:
Check the quadrant before writing the answer. 330° is in the fourth quadrant, where sine is negative, so the answer is $-\frac{1}{2}$, not $\frac{1}{2}$.
Leaving the calculator in radian mode.
Where it slips in:
A student types "sin(330)" expecting degrees, but the calculator is set to radians and returns roughly $-0.132$, which looks plausible and goes unquestioned.
Don't do this:
Do not trust a decimal without checking the angle unit. The number $-0.132$ is $\sin$ of 330 radians, a completely different angle.
The correct way:
Set the mode to degrees for a degree question, or convert first: $330^\circ = \frac{11\pi}{6}$, and $\sin\frac{11\pi}{6} = -\frac{1}{2}$.
Using the wrong reference angle.
Where it slips in:
A student subtracts from the wrong axis, computing $330^\circ - 270^\circ = 60^\circ$ and using $\sin 60^\circ$ instead of $\sin 30^\circ$.
Don't do this:
Do not measure the reference angle from a vertical axis. For an angle near a full turn, the reference is the gap to the horizontal axis at 360°.
The correct way:
Use $360^\circ - 330^\circ = 30^\circ$. The reference angle is always the acute gap to the nearest point on the $x$-axis, which is 0° or 360° here.
Confusing sin 330 degrees with cos 330 degrees.
Where it slips in:
A student swaps the coordinates and reports $\frac{\sqrt{3}}{2}$, which is actually the cosine (the $x$-coordinate), not the sine.
Don't do this:
Do not read the horizontal coordinate for sine. Sine is the $y$-coordinate, cosine is the $x$-coordinate.
The correct way:
Read the vertical coordinate for sine: $\sin 330^\circ = -\frac{1}{2}$, while $\cos 330^\circ = \frac{\sqrt{3}}{2}$. They are different values from the same point.
Practice Problems On Sin 330 Degrees
Work each one with the reference-angle-then-sign method. Answers follow each line.
Write $\sin 330^\circ$ as a fraction and as a decimal.
(Answer: $-\frac{1}{2}$ and $-0.5$.)Find $\cos 330^\circ$ using the same unit-circle point.
(Answer: $\frac{\sqrt{3}}{2} \approx 0.8660$, the $x$-coordinate.)Find $\tan 330^\circ$.
(Answer: $\dfrac{\sin 330^\circ}{\cos 330^\circ} = \dfrac{-1/2}{\sqrt{3}/2} = -\dfrac{1}{\sqrt{3}} \approx -0.5774$.)Evaluate $\sin\dfrac{11\pi}{6}$.
(Answer: $-\frac{1}{2}$, the same angle in radians.)Which other angle in $[0^\circ, 360^\circ)$ has the same sine as 330°?
(Answer: $210^\circ$, also $-\frac{1}{2}$, since both share the 30° reference angle below the axis.)A Ferris wheel of radius 20 m has its centre 22 m above the ground. Find a rider's height at 330°.
(Answer: $22 + 20\sin 330^\circ = 22 + 20(-\tfrac{1}{2}) = 12$ m.)
Where Should You Go Next After Sin 330 Degrees?
Sin 330 degrees is one point on the unit circle, and a few natural doors open from here.
Sin 30 degrees. Master the reference angle that powers 330°, and the whole 30°-family becomes automatic.
Unit circle with tangent. See how sine, cosine, and tangent all live on one diagram, so any angle's value is a coordinate you can read off.
What is a radian. Understand why 330° is $\frac{11\pi}{6}$, and radian arguments stop looking strange.
If your child is building these foundations, a live Bhanzu trainer teaches the unit circle starting from the "why" (sine as a height, not a formula to memorise) in the Bhanzu trigonometry program.
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