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

What Is The Value Of Cos 11pi/6?

Cos 11pi/6 is $\frac{\sqrt{3}}{2}$, or about $0.8660$ to four decimal places. Written in both common forms of the angle, $\cos\frac{11\pi}{6} = \cos 330^\circ = \frac{\sqrt{3}}{2}$. The value is positive, and it is exactly the same size as $\cos 30^\circ$, because $330^\circ$ and $30^\circ$ share a reference angle.

Here is the angle in both radians and degrees, side by side:

$$\frac{11\pi}{6} \text{ radians} = 330^\circ, \qquad \cos\frac{11\pi}{6} = \frac{\sqrt{3}}{2} \approx 0.8660$$

The radian form comes from the conversion $\pi \text{ radians} = 180^\circ$, so $\frac{11\pi}{6} = \frac{11 \times 180^\circ}{6} = 330^\circ$. If the idea of measuring an angle in $\pi$ is new, what is a radian explains where the unit comes from.

How Do You Find Cos 11pi/6?

Finding cos 11pi/6 takes two decisions: which quadrant the angle sits in, and what its reference angle is. Get those two right and the value follows.

Step 1: Locate the quadrant. A full turn is $360^\circ$, split into four quadrants of $90^\circ$ each. Since $330^\circ$ is between $270^\circ$ and $360^\circ$, the angle lands in Quadrant IV.

Step 2: Find the reference angle. The reference angle is the acute angle between the terminal arm and the horizontal axis. In Quadrant IV it is $360^\circ - 330^\circ = 30^\circ$, which is $\frac{\pi}{6}$ in radians.

Step 3: Apply the sign for that quadrant. Use the CAST rule (also called ASTC), which records where each ratio is positive:

• A (Quadrant I): all ratios positive.

• S (Quadrant II): only sine positive.

• T (Quadrant III): only tangent positive.

• C (Quadrant IV): only cosine positive.

Quadrant IV is the C quadrant, so cosine is positive there. That gives:

$$\cos\frac{11\pi}{6} = +\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2} \approx 0.8660$$

The reference-angle method works for every special angle. To see all of them in one place, the trigonometric table lists sine, cosine, and tangent for the standard angles, and trigonometric ratios in radians shows the same values indexed by their radian measure.

Where Does 11pi/6 Sit On The Unit Circle?

On the unit circle (radius $1$, centred at the origin), the angle $\frac{11\pi}{6}$ is measured anticlockwise from the positive $x$-axis and stops just short of a full turn, at the point $\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$. Cosine is defined as the $x$-coordinate of that point, so reading it straight off the circle gives:

$$\cos\frac{11\pi}{6} = x\text{-coordinate} = \frac{\sqrt{3}}{2}$$

The $y$-coordinate is $-\frac{1}{2}$, which is why $\sin\frac{11\pi}{6} = -\frac{1}{2}$: below the horizontal axis, sine is negative, while the $x$-coordinate stays to the right of the vertical axis, so cosine stays positive.

Reading the value two ways, from the right triangle and from the unit circle, is worth doing on purpose. The reference angle $\frac{\pi}{6}$ builds a 30-60-90 right triangle whose adjacent-over-hypotenuse ratio is $\frac{\sqrt{3}}{2}$, and the unit circle gives the same $\frac{\sqrt{3}}{2}$ as a coordinate. A short refresher on the three core ratios lives at sin cos tan.

How Do You Prove Cos 11pi/6 Using Identities?

The reference angle is the fastest route, but two standard identities confirm the same value and are useful when a question demands an algebraic derivation.

Method 1: The even and periodic property. The angle $\frac{11\pi}{6}$ is one step short of a full turn, since $\frac{11\pi}{6} = 2\pi - \frac{\pi}{6}$. Cosine satisfies $\cos(2\pi - \theta) = \cos\theta$, so:

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

Method 2: The angle-sum route. Write $\frac{11\pi}{6}$ as $\frac{3\pi}{2} + \frac{\pi}{3}$, then use $\cos\left(\frac{3\pi}{2} + \theta\right) = \sin\theta$:

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

That second line also exposes a neat co-function fact: $\cos\frac{11\pi}{6}$ equals $\sin\frac{\pi}{3}$, which is $\sin 60^\circ$. Cosine and sine trade places across complementary angles, a relationship set out in full at cofunction identities.

What Are The Cosine Values Of Related Angles?

Placing cos 11pi/6 next to its neighbours makes the reference-angle pattern visible. Each Quadrant IV angle below borrows the size of a first-quadrant cosine and keeps it positive.

Table: Cosine of angles related to 11π/6, in radians and degrees.

Read down the Quadrant IV rows and the cosines are all positive, climbing back toward $1$ as the angle nears a full turn. Read the Quadrant II rows and they are negative. The sign is doing the same job in every row: it records which side of the vertical axis the point sits on.

Why Is Cos 11pi/6 Positive?

The sign is not a rule to memorise on its own. It comes straight from the geometry of the circle.

• Cosine is a horizontal position. On the unit circle, cosine is the $x$-coordinate. Anything to the right of the vertical axis has a positive $x$-coordinate, and anything to the left has a negative one.

• Quadrant IV is on the right. The angle $330^\circ$ ends in the lower-right quadrant, so its point sits to the right of centre. A right-of-centre point has a positive $x$-coordinate, so its cosine is positive.

• The size copies the reference angle. The reference angle $\frac{\pi}{6}$ fixes how far from the axis the point is, giving the magnitude $\frac{\sqrt{3}}{2}$. The quadrant fixes only the sign, not the size.

Put together, the point at $330^\circ$ is close to the positive $x$-axis, low down and far to the right, so its horizontal distance is large and positive. That is the whole reason cos 11pi/6 comes out as a positive $\frac{\sqrt{3}}{2}$ rather than a negative number.

Who Discovered Cosine Values Like Cos 11pi/6?

Tables of these values existed long before calculators, and long before the modern words for the ratios. Mathematicians across Greece, India, and the Islamic world spent centuries computing them by hand.

Two earlier figures built the ground Madhava stood on:

• Hipparchus of Nicaea (c. 190 – c. 120 BCE, Greece) compiled the first known trigonometric table, a table of chords, around 150 BCE, which is why he is often called the father of trigonometry.

• Aryabhata (476 – 550 CE, India) tabulated the sine function (which he called jya) in his Aryabhatiya around 500 CE, giving values at regular intervals that later reached the Islamic world and Europe.

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

A cosine near the end of a full turn is not just an exam value. The same $\frac{\sqrt{3}}{2}$ shows up wherever something rotates or oscillates.

• Alternating current: the voltage in a mains socket rises and falls as a cosine wave, and engineers read off values at specific phase angles like $330^\circ$ to find the instantaneous voltage during a cycle.

• Circular motion: a point moving around a wheel or an orbit has a horizontal position given by cosine, so cos 11pi/6 is the sideways position after turning $330^\circ$.

• Navigation and GPS: bearings and satellite positions resolve into horizontal and vertical components using cosine and sine of the angle, and Quadrant IV angles are routine.

• Computer graphics: rotating a sprite or a 3D model by $330^\circ$ multiplies its coordinates by cosine and sine of that angle, so this exact value is computed millions of times a second in games.

• Sound and signals: any pure tone is a cosine wave, and its value at a given phase is read the same way as cos 11pi/6.

One value, $\frac{\sqrt{3}}{2}$, sits behind the power grid, the arcade, and the satellite fix. The mathematics of turning is the same wherever the turning happens.

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

These three errors account for most wrong answers on this angle, confirmed against the reference-angle and quadrant-sign questions students ask most often about $\frac{11\pi}{6}$.

Making cos 11pi/6 negative.

Where it slips in:

A student sees an angle past $180^\circ$ and assumes the cosine must be negative, as it is in Quadrants II and III.

Don't do this:

Do not attach a minus sign by habit. Quadrant IV is the one place past $180^\circ$ where cosine is positive.

The correct way:

Use CAST. Quadrant IV is the C quadrant, so cosine is positive: $\cos\frac{11\pi}{6} = +\frac{\sqrt{3}}{2}$.

Miscomputing the reference angle.

Where it slips in:

A student subtracts from the wrong number, using $330^\circ - 270^\circ = 60^\circ$ or $330^\circ - 180^\circ = 150^\circ$ instead of measuring to the nearest horizontal axis.

Don't do this:

Do not subtract from $270^\circ$ or $180^\circ$ for a Quadrant IV angle.

The correct way:

In Quadrant IV the reference angle is $360^\circ - \theta$, so $360^\circ - 330^\circ = 30^\circ = \frac{\pi}{6}$, which gives the magnitude $\frac{\sqrt{3}}{2}$.

Leaving the calculator in degree mode.

Where it slips in:

A student types $\cos(11\pi/6)$ with the calculator set to degrees, so it reads $\frac{11\pi}{6} \approx 5.76$ as $5.76^\circ$ and returns roughly $0.9949$.

Don't do this:

Do not enter a radian angle while the mode is set to degrees.

The correct way:

Switch the calculator to radian mode before entering $\frac{11\pi}{6}$, or convert to $330^\circ$ first and stay in degree mode. Either way the answer is $\frac{\sqrt{3}}{2} \approx 0.8660$.

Practice Problems On Cos 11pi/6

Work each one with the reference-angle method, then check the answer that follows.

1. State $\cos\frac{11\pi}{6}$ as an exact value and as a decimal to four places.
   (Answer: $\frac{\sqrt{3}}{2} \approx 0.8660$.)

2. Find $\sin\frac{11\pi}{6}$ using the same point on the unit circle.
   (Answer: $-\frac{1}{2}$, the $y$-coordinate at $330^\circ$.)

3. Find $\tan\frac{11\pi}{6}$.
   (Answer: $\dfrac{\sin}{\cos} = \dfrac{-1/2}{\sqrt{3}/2} = -\dfrac{1}{\sqrt{3}} = -\dfrac{\sqrt{3}}{3} \approx -0.5774$.)

4. Find $\sec\frac{11\pi}{6}$, the reciprocal of cosine.
   (Answer: $\dfrac{1}{\sqrt{3}/2} = \dfrac{2}{\sqrt{3}} = \dfrac{2\sqrt{3}}{3} \approx 1.1547$.)

5. Which other special angle between $0$ and $2\pi$ has the same cosine as $\frac{11\pi}{6}$?
   (Answer: $\frac{\pi}{6}$, since both have reference angle $\frac{\pi}{6}$ and lie where cosine is positive.)

6. Convert $\frac{11\pi}{6}$ to degrees and confirm the quadrant.
   (Answer: $330^\circ$, Quadrant IV.)

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

One special value opens onto the whole system of angles and ratios. Three natural doors lead on from here.

1. The trigonometric table. See every special-angle value for sine, cosine, and tangent in one grid, so any angle like $\frac{11\pi}{6}$ becomes a quick lookup.

2. The unit circle with tangent. Watch how each point's coordinates give cosine, sine, and tangent together, which is the picture behind every value on this page.

3. What is a radian. Get comfortable with measuring angles in $\pi$, so $\frac{11\pi}{6}$ reads as naturally as $330^\circ$.

If your child is building these foundations, a live Bhanzu trainer teaches angles from the unit circle outward, starting with why each value looks the way it does, in the Bhanzu trigonometry program.