Cos 240 Degrees: Exact Value −1/2 Explained

What Is The Value Of Cos 240 Degrees?

Cos 240 degrees is equal to $-\dfrac{1}{2}$, or $-0.5$ in decimal form. Written with the angle in radians, the same fact reads $\cos\frac{4\pi}{3} = -\frac{1}{2}$, because $240^\circ$ and $\frac{4\pi}{3}$ are the same angle measured two ways.

The value is negative for one reason: $240^\circ$ sits in Quadrant III, and cosine is negative everywhere in that quadrant. The size of the number, $\frac{1}{2}$, comes from the reference angle $60^\circ$, whose cosine is $\frac{1}{2}$.

$$\cos 240^\circ = \cos\frac{4\pi}{3} = -\frac{1}{2} = -0.5$$

How Do You Find Cos 240 Degrees Using The Reference Angle?

The reference-angle method is the fastest route, and it works for any angle. It has two parts: find the size, then fix the sign.

• Find the reference angle. Since $240^\circ$ is between $180^\circ$ and $270^\circ$, it lies in Quadrant III, and the reference angle is $240^\circ - 180^\circ = 60^\circ$.

• Take the cosine of the reference angle. From the special angles, $\cos 60^\circ = \frac{1}{2}$.

• Fix the sign by the quadrant. In Quadrant III both $x$ and $y$ are negative, so cosine (an $x$-reading) is negative.

Putting the three steps together:

$$\cos 240^\circ = -\cos 60^\circ = -\frac{1}{2}$$

A memory aid for the sign is ASTC ("All, Sine, Tangent, Cosine"), naming which ratio stays positive in each quadrant from I to IV. Quadrant III is the "Tangent" quadrant, so only tangent is positive there, and cosine is negative. For a full walk-through of this idea, see reference angle and the trigonometric ratios of specific angles.

Where Does 240 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 angle's ray meets the circle has coordinates $(\cos\theta, \sin\theta)$. Cosine is always the $x$-coordinate.

At $240^\circ$ the point is $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$. The $x$-coordinate is $-\frac{1}{2}$, which is exactly cos 240 degrees, and it is negative because the point sits to the left of the vertical axis.

$$240^\circ = \frac{4\pi}{3}, \qquad P = \left(-\tfrac{1}{2},, -\tfrac{\sqrt{3}}{2}\right), \qquad \cos 240^\circ = -\tfrac{1}{2}$$

How Do You Derive Cos 240 Degrees With The Angle-Sum Formula?

When you want the exact value without a picture, an angle-sum (or angle-difference) identity gives it cleanly. Write $240^\circ$ as a sum of angles whose sine and cosine you already know.

Take $240^\circ = 180^\circ + 60^\circ$ and apply $\cos(A + B) = \cos A \cos B - \sin A \sin B$:

$$\cos 240^\circ = \cos(180^\circ + 60^\circ)$$

$$= \cos 180^\circ \cos 60^\circ - \sin 180^\circ \sin 60^\circ$$

$$= (-1)\left(\tfrac{1}{2}\right) - (0)\left(\tfrac{\sqrt{3}}{2}\right)$$

$$= -\tfrac{1}{2}$$

A second decomposition checks the answer. Using $240^\circ = 270^\circ - 30^\circ$ and $\cos(A - B) = \cos A \cos B + \sin A \sin B$:

$$\cos 240^\circ = \cos 270^\circ \cos 30^\circ + \sin 270^\circ \sin 30^\circ = (0)\left(\tfrac{\sqrt{3}}{2}\right) + (-1)\left(\tfrac{1}{2}\right) = -\tfrac{1}{2}$$

Both routes agree: $\cos 240^\circ = -\frac{1}{2}$. The same building blocks appear in cos 180 degrees and cos 60 degrees.

What Are The Other Trigonometric Ratios At 240 Degrees?

Once cosine and sine are known, the remaining four ratios follow from their definitions. At $240^\circ$ the unit-circle point is $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$, so $\cos 240^\circ = -\frac{1}{2}$ and $\sin 240^\circ = -\frac{\sqrt{3}}{2}$.

Table: The six trigonometric ratios evaluated at 240 degrees (4π/3 radians), exact and to 4 decimal places.

Notice that tangent and cotangent are positive at $240^\circ$: they are ratios of two negatives, so the signs cancel. That is exactly what ASTC predicts for Quadrant III. For the definitions behind these ratios, see sin cos tan and the full trigonometric table.

How Does Cos 240 Degrees Compare To Nearby Angles?

Cos 240 degrees belongs to a family of "reference-angle 60°" cosines whose values differ only in sign. Seeing them together makes the pattern obvious.

Table: Cosine values for angles sharing a 60° or 30° reference, in degrees and radians.

Cos 240 degrees shares its exact magnitude with cos 120 degrees (both $-\frac{1}{2}$) because $120^\circ$ and $240^\circ$ are mirror images across the $x$-axis pair. Compare it with the radian-labelled pages cos 2pi/3 and cos 5pi/3 to see the same values written the other way.

Why Is Cos 240 Degrees Negative?

The sign is not a rule to memorise; it is geometry you can see. Cosine reads the horizontal position of the point on the unit circle, and where that point sits decides the sign.

• The point is on the left. At $240^\circ$ the ray has swung past the vertical axis into Quadrant III, so the point lies to the left of centre. A position left of centre is a negative $x$-value, and cosine is that $x$-value.

• Quadrant III makes both coordinates negative. Below and to the left of the origin, $x < 0$ and $y < 0$, so cosine and sine are both negative there. That is why $\cos 240^\circ$ and $\sin 240^\circ$ are each below zero.

• The reference angle sets the size. The ray makes a $60^\circ$ angle with the negative $x$-axis, and $\cos 60^\circ = \frac{1}{2}$, so the horizontal reach is $\frac{1}{2}$ in length, carried with a minus sign.

Put together, the horizontal coordinate is $-\frac{1}{2}$, and that is cos 240 degrees. A stronger grip on the horizontal-and-vertical reading comes from the unit circle with tangent.

Who Discovered The Angles Behind Cos 240 Degrees?

Nobody "discovered" the number $-\frac{1}{2}$ on its own. What people built, over centuries, were the tables of chord and sine values that let anyone read a trigonometric ratio for any angle, the ancestors of the unit circle we use today.

Two earlier figures shaped the same story:

• Hipparchus of Nicaea (c. 190–120 BCE, Greece) is often called the founder of trigonometry for compiling the first known table of chords, the direct predecessor of the sine table.

• Claudius Ptolemy (c. 100–170 CE, Roman Egypt) refined chord tables in his Almagest, tabulating values precise enough to drive astronomy for over a thousand years.

Where Is Cos 240 Degrees Used In The Real World?

A single cosine value rarely appears alone in daily life, but the function that produces it runs a surprising amount of technology, and angles past $180^\circ$ like $240^\circ$ are ordinary inside it.

• Alternating current: household electricity is a cosine wave, and a three-phase supply spaces its phases exactly $120^\circ$ apart, so one phase sits at $240^\circ$ while another sits at $120^\circ$.

• Sound and vibration: speakers, strings, and springs oscillate as cosine curves, and the value at $240^\circ$ marks a specific point in each cycle.

• GPS and navigation: position fixes rely on the phase of repeating signals, phases measured as angles right around the circle.

• Computer graphics: rotating a game character or a 3D model applies cosine and sine of the turn angle, and a $240^\circ$ rotation is a routine input.

• Astronomy: orbital and rotational positions are tracked with the very angle tables that trigonometry began as.

Wherever something repeats in a smooth cycle, cosine describes it, and $240^\circ$ is simply one place on that cycle.

What Are The Most Common Mistakes With Cos 240 Degrees?

These four slips account for most wrong answers on this angle. Each is easy to avoid once you name it.

Dropping the negative sign.

Where it slips in:

A student finds the reference angle, writes $\cos 60^\circ = \frac{1}{2}$, and stops there, forgetting the quadrant.

Don't do this:

Do not report $\cos 240^\circ = \frac{1}{2}$. That is the size without the sign.

The correct way:

Check the quadrant before writing the answer. $240^\circ$ is in Quadrant III where cosine is negative, so $\cos 240^\circ = -\frac{1}{2}$.

Leaving the calculator in the wrong mode.

Where it slips in:

A student types "cos(240)" with the calculator set to radians and reads off roughly $0.9998$, which is the cosine of $240$ radians, not $240$ degrees.

Don't do this:

Do not trust the display until the angle mode matches the angle. Radian mode on a degree question gives a completely different number.

The correct way:

Set the calculator to DEG for $240^\circ$, or convert first: $240^\circ = \frac{4\pi}{3}$ and use RAD. Either way the answer is $-0.5$.

Taking the wrong reference angle.

Where it slips in:

A student subtracts from the nearest axis incorrectly, using $240^\circ - 90^\circ = 150^\circ$ or $270^\circ - 240^\circ = 30^\circ$, and evaluates the wrong angle.

Don't do this:

Do not measure the reference angle from the vertical axis. For a Quadrant III angle the reference is taken from the horizontal ($180^\circ$) axis.

The correct way:

Use $240^\circ - 180^\circ = 60^\circ$. The reference angle is the acute angle to the nearest part of the $x$-axis, which is $60^\circ$.

Swapping cosine and sine.

Where it slips in:

A student reads the $y$-coordinate of the unit-circle point instead of the $x$-coordinate and reports $-\frac{\sqrt{3}}{2}$.

Don't do this:

Do not give $\cos 240^\circ = -\frac{\sqrt{3}}{2}$. That value is $\sin 240^\circ$, the vertical coordinate.

The correct way:

Remember cosine is the horizontal ($x$) coordinate. At $240^\circ$ that is $-\frac{1}{2}$, while the vertical $-\frac{\sqrt{3}}{2}$ is the sine.

Practice Problems On Cos 240 Degrees

Work each one, then check the answer that follows.

1. Convert $240^\circ$ to radians.
   (Answer: $240 \times \frac{\pi}{180} = \frac{4\pi}{3}$.)

2. State the reference angle of $240^\circ$ and its quadrant.
   (Answer: reference angle $60^\circ$, Quadrant III.)

3. Evaluate $\sec 240^\circ$.
   (Answer: $\sec 240^\circ = \frac{1}{\cos 240^\circ} = \frac{1}{-1/2} = -2$.)

4. Find $\cos 240^\circ + \cos 120^\circ$.
   (Answer: $-\frac{1}{2} + \left(-\frac{1}{2}\right) = -1$.)

5. Use $240^\circ = 180^\circ + 60^\circ$ to derive $\cos 240^\circ$.
   (Answer: $\cos180^\circ\cos60^\circ - \sin180^\circ\sin60^\circ = -\frac{1}{2}$.)

6. Evaluate $2\cos^2 240^\circ - 1$.
   (Answer: $2\left(\frac{1}{4}\right) - 1 = -\frac{1}{2}$, which is $\cos 480^\circ$.)

Where Should You Go Next After Cos 240 Degrees?

Cos 240 degrees is one point on a much larger map, and a few natural doors open from here.

1. Trigonometric ratios of specific angles. Learn the full set of exact values for the standard angles, the toolkit every one of these problems draws on.

2. Sin 240 degrees. The companion value at the same angle, worked the same way, so you can see cosine and sine side by side.

3. Cosine function. Step back to the whole function, its graph, and how a single value fits the repeating wave.

If your child is building these foundations, a live Bhanzu trainer teaches trigonometry from the unit circle up, so values like cos 240 degrees are understood, not memorised, in the Bhanzu trigonometry program.