Sin 11pi/6: Exact Value On The Unit Circle

What Is The Value Of Sin 11pi/6?

Sin 11pi/6 is $-\dfrac{1}{2}$, or $-0.5$ in decimal form. Written with the angle in both units, $\sin\frac{11\pi}{6} = \sin 330^\circ = -\frac{1}{2}$.

The angle here is measured in radians. One full turn around a circle is $2\pi$ radians, which equals $360^\circ$, so each radian is a slice of that turn. If radians still feel new, start with what is a radian and the wider trigonometric ratios in radians.

To convert the angle, multiply by $\frac{180^\circ}{\pi}$:

$$\frac{11\pi}{6} \times \frac{180^\circ}{\pi} = \frac{11 \times 180^\circ}{6} = 11 \times 30^\circ = 330^\circ$$

So $\frac{11\pi}{6}$ and $330^\circ$ are the same angle. Both name a point almost all the way around the circle, sitting just short of a full turn.

How Do You Find Sin 11pi/6?

Finding the sine of any special angle past $90^\circ$ takes three quick decisions: which quadrant, what reference angle, and what sign.

• Quadrant. $330^\circ$ is between $270^\circ$ and $360^\circ$, so the angle sits in the fourth quadrant.

• Reference angle. The reference angle is the gap to the nearest horizontal axis. Here that is $360^\circ - 330^\circ = 30^\circ$, or in radians $2\pi - \frac{11\pi}{6} = \frac{\pi}{6}$.

• Sign. Sine tracks the vertical direction. In the fourth quadrant the vertical direction points down, so sine is negative.

Put the three together. The reference angle gives the size, $\sin 30^\circ = \frac{1}{2}$, and the fourth quadrant flips the sign:

$$\sin\frac{11\pi}{6} = -\sin\frac{\pi}{6} = -\frac{1}{2}$$

A memory aid many students use is ASTC (sometimes read as CAST): in quadrants one to four, the functions that stay positive are All, then Sine, then Tangent, then Cosine. The fourth quadrant is the "C" corner, where only cosine is positive, so sine and tangent come out negative there. You can check the size of the reference value any time against the trigonometric table or the sin cos tan overview.

Where Does 330 Sit On The Unit Circle?

The unit circle is a circle of radius $1$ centred at the origin. For any angle measured from the positive $x$-axis, the point where the circle is cut has coordinates $(\cos\theta, \sin\theta)$. The sine is simply the $y$-coordinate, and the cosine is the $x$-coordinate.

At $\frac{11\pi}{6}$ (that is $330^\circ$), the point sits low and to the right, just below the positive $x$-axis. Its coordinates are:

$$\left(\cos\frac{11\pi}{6},\ \sin\frac{11\pi}{6}\right) = \left(\frac{\sqrt{3}}{2},\ -\frac{1}{2}\right)$$

The $x$-coordinate $\frac{\sqrt{3}}{2} \approx 0.866$ is positive, which is why cosine is positive in the fourth quadrant. The $y$-coordinate $-\frac{1}{2}$ is below the axis, which is exactly why the sine is negative. Reading a value straight off the picture is often faster than any rule, so a labelled circle like unit circle with tangent is worth keeping close.

Can Sin 11pi/6 Be Written As An Exact Fraction?

Yes. Because $\frac{\pi}{6}$ is one of the special angles, the value is a clean exact fraction, not a rounded decimal. Two short routes both land on the same answer.

The reference-angle route uses the fact that reflecting an angle into the fourth quadrant keeps the size and flips the sign of the sine:

$$\sin\frac{11\pi}{6} = \sin\left(2\pi - \frac{\pi}{6}\right) = -\sin\frac{\pi}{6} = -\frac{1}{2}$$

The co-function route rewrites the reference sine as a cosine, since $\sin 30^\circ = \cos 60^\circ$:

$$\sin\frac{11\pi}{6} = -\sin\frac{\pi}{6} = -\cos\frac{\pi}{3} = -\frac{1}{2}$$

Both confirm the exact value $-\frac{1}{2}$. The reference value $\sin\frac{\pi}{6}$ is worked in full at sin pi/6 and in degree form at sin 30 degrees, and the rewriting rules live in cofunction identities.

You can also anchor the same $\frac{1}{2}$ in a right triangle. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, the side opposite the $30^\circ$ angle is half the hypotenuse, so $\sin 30^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2}$. The unit circle then carries that ratio around to the fourth quadrant and attaches the negative sign.

Table 1: Sine, cosine, and tangent for the reference-angle π/6 family across the four quadrants.

Every angle in this family shares the same reference angle $\frac{\pi}{6}$, so the size of each ratio is identical. Only the signs change, and the signs are set entirely by the quadrant.

Why Is Sin 11pi/6 Negative?

The minus sign is not a rule to memorise. It falls straight out of the geometry once you read sine as a height.

• Sine is a vertical measurement. On the unit circle, $\sin\theta$ is the $y$-coordinate of the point. Above the $x$-axis the height is positive; below it, the height is negative.

• The fourth quadrant is below the axis. The point for $330^\circ$ sits just under the positive $x$-axis, so its $y$-coordinate is below zero. A below-zero height means a negative sine.

• The size still comes from $30^\circ$. The point is a mirror image of the $30^\circ$ point across the $x$-axis, so the height has the same magnitude, $\frac{1}{2}$, and only the direction is reversed.

That is the whole reason $\sin\frac{11\pi}{6} = -\frac{1}{2}$ rather than $+\frac{1}{2}$. Cosine, being a horizontal measurement, stays positive at $330^\circ$ because the point is still to the right of the $y$-axis. This angle turns up in school syllabuses on both sides of the world, from India's NCERT Class 11 trigonometry to the United States Common Core high-school functions strand, so the reasoning is worth owning once and reusing everywhere.

Who Shaped Our Understanding Of The Sine Function?

The sine did not begin as a wave or a unit-circle coordinate. It began as the length of a half-chord in a table, built by astronomers who needed to predict where the stars would be.

Two other figures shaped the same idea:

• Hipparchus of Nicaea (c. 190–120 BCE, Greece) built one of the earliest known tables of chords, the direct ancestor of the sine table, to do astronomy.

• Al-Battani (c. 858–929 CE, in what is now Turkey and Iraq) worked with sines rather than chords and sharpened the trigonometry later used across the medieval world.

Where Is Sin 11pi/6 Used In The Real World?

A single sine value looks abstract, but negative sines model anything that dips below a resting line, and $330^\circ$ is a common sampling point in those cycles.

• Ferris wheels and rotating rides. The height of a seat is a sine of the rotation angle. Past the bottom of the turn, at angles like $330^\circ$, the seat is below the hub, so the height reads negative.

• Alternating current. Household electricity is a sine wave. At the part of the cycle matching $330^\circ$, the voltage is negative, meaning the current is pushing the other way.

• Tides and sound. Tide height and the pressure in a sound wave both rise and fall as sines, dropping below their average level exactly where the sine is negative.

• Computer graphics. Rotating a point around a circle uses $\sin$ and $\cos$ of the angle, and a fourth-quadrant angle like $\frac{11\pi}{6}$ places the point in the lower-right of the turn.

One value, $-\frac{1}{2}$, quietly describes a wheel seat, a wall socket, a wave, and a spinning game sprite. The same math shows up wherever something cycles.

What Are The Most Common Mistakes With Sin 11pi/6?

These four errors account for most lost marks on fourth-quadrant special angles, matching the reference-angle and quadrant prompts that competitor pages and student-question boards surface again and again.

Giving the answer a positive sign.

Where it slips in:

A student finds the reference value $\sin 30^\circ = \frac{1}{2}$ and writes that as the final answer, forgetting the quadrant.

Don't do this:

Do not stop at the reference value. The size is only half the job.

The correct way:

Read the quadrant first. $\frac{11\pi}{6}$ is in the fourth quadrant, where sine is negative, so the answer is $-\frac{1}{2}$, not $+\frac{1}{2}$.

Leaving the calculator in degree mode.

Where it slips in:

A student types $\sin(11\pi/6)$ with the calculator set to degrees, so it computes the sine of about $5.76^\circ$ and returns roughly $0.1$.

Don't do this:

Do not enter a radian angle while the mode reads DEG.

The correct way:

Switch the calculator to radian mode for $\frac{11\pi}{6}$, or convert to $330^\circ$ first and stay in degree mode. Both give $-0.5$.

Using the wrong reference angle.

Where it slips in:

A student subtracts from the wrong axis, using $330^\circ - 270^\circ = 60^\circ$ instead of the gap to $360^\circ$.

Don't do this:

Do not measure the reference angle from the vertical axis for a fourth-quadrant angle.

The correct way:

In the fourth quadrant, the reference angle is $360^\circ - \theta$. Here $360^\circ - 330^\circ = 30^\circ$, giving reference value $\frac{1}{2}$.

Confusing the angle with its coterminal negative.

Where it slips in:

A student rewrites $\frac{11\pi}{6}$ as $-\frac{\pi}{6}$ and then treats the two as different problems with different answers.

Don't do this:

Do not expect a different value. They are the same position on the circle.

The correct way:

$\frac{11\pi}{6}$ and $-\frac{\pi}{6}$ are coterminal, so $\sin\frac{11\pi}{6} = \sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2}$. See coterminal angles for why adding or subtracting $2\pi$ never changes a trig value.

Practice Problems On Sin 11pi/6

Work each one, then check against the answer that follows.

1. Convert $\frac{11\pi}{6}$ to degrees.
   (Answer: $330^\circ$.)

2. State the reference angle of $\frac{11\pi}{6}$.
   (Answer: $\frac{\pi}{6}$, or $30^\circ$.)

3. Find $\sin\frac{11\pi}{6}$.
   (Answer: $-\frac{1}{2}$.)

4. Find $\cos\frac{11\pi}{6}$.
   (Answer: $\frac{\sqrt{3}}{2}$.)

5. Find $\tan\frac{11\pi}{6}$.
   (Answer: $\dfrac{\sin}{\cos} = \dfrac{-1/2}{\sqrt{3}/2} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$.)

6. Evaluate $\sin\frac{11\pi}{6} + \sin\frac{\pi}{6}$.
   (Answer: $-\frac{1}{2} + \frac{1}{2} = 0$.)

Where Should You Go Next After Sin 11pi/6?

This one value opens several natural doors into the rest of trigonometry.

1. Trigonometric ratios in radians. Read the whole circle in radians, so angles like $\frac{2\pi}{3}$ and $\frac{3\pi}{2}$ feel as familiar as degrees.

2. Sin 2pi/3 and sin 3pi/2. Practise the same reference-angle and quadrant method on other special angles.

3. Cofunction identities. See why every sine can be rewritten as a cosine, the trick used in the exact-value section above.

If your child is building these foundations, a live Bhanzu trainer teaches special angles starting from the unit circle, so the signs make sense instead of being memorised, in the Bhanzu trigonometry program.