Cos 150 Degrees: Exact Value & Unit Circle

What Is The Value Of Cos 150 Degrees?

Cos 150 Degrees is $-\frac{\sqrt{3}}{2}$, or about $-0.8660$ rounded to four decimal places. The angle can be written two ways, $150^\circ$ in degrees and $\frac{5\pi}{6}$ in radians, and both name the same rotation.

$$\cos 150^\circ = \cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2} \approx -0.8660$$

The value is exact in surd form. The decimal $-0.8660$ is only a rounding of $-\frac{\sqrt{3}}{2}$, so the fraction with the square root is the form to quote in any proof or exam answer. The minus sign is not optional, and the next section explains where it comes from.

How Do You Find Cos 150 Degrees?

The fastest route uses the reference angle and the quadrant sign. Every angle has a reference angle, the acute angle it makes with the horizontal axis, and the cosine of the original angle is the cosine of that reference angle with a sign fixed by the quadrant.

Follow three steps.

• Locate the quadrant. $150^\circ$ is between $90^\circ$ and $180^\circ$, so it lands in the second quadrant.

• Find the reference angle. In the second quadrant the reference angle is $180^\circ - 150^\circ = 30^\circ$.

• Fix the sign. Using the ASTC rule (All, Sine, Tangent, Cosine positive by quadrant), only sine is positive in the second quadrant, so cosine is negative there.

Putting those together gives the supplementary-angle identity:

$$\cos 150^\circ = \cos(180^\circ - 30^\circ) = -\cos 30^\circ = -\frac{\sqrt{3}}{2}$$

Because $\cos 30^\circ$ is a value most students already know, the whole problem reduces to remembering one special angle and one sign rule. For the full set of these standard values, see the trigonometric ratios of specific angles.

Can You Derive Cos 150 Degrees With The Angle-Sum Formula?

Yes, and it confirms the same answer without leaning on the reference angle. Write $150^\circ$ as $90^\circ + 60^\circ$ and apply the cosine addition formula $\cos(A+B) = \cos A \cos B - \sin A \sin B$:

$$\cos(90^\circ + 60^\circ) = \cos 90^\circ \cos 60^\circ - \sin 90^\circ \sin 60^\circ$$

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

There is also a co-function route. Since $\cos\theta = \sin(90^\circ - \theta)$, you get $\cos 150^\circ = \sin(90^\circ - 150^\circ) = \sin(-60^\circ) = -\sin 60^\circ = -\frac{\sqrt{3}}{2}$. Three independent methods, one answer, which is exactly what a correct exact value should do. The complementary-angle relationships are what make that last route work.

Where Does 150 Degrees Sit On The Unit Circle?

On the unit circle, the point for an angle is $(\cos\theta, \sin\theta)$, so the cosine is simply the horizontal coordinate of that point. For $150^\circ$, the point is $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, which is about $(-0.8660, 0.5)$.

$$\left(\cos 150^\circ, \sin 150^\circ\right) = \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$$

The $x$-coordinate is negative because the point lies to the left of the vertical axis, in the second quadrant. That is the geometric reason the sign flips: past $90^\circ$, the terminal side of the angle crosses to the left half of the circle, and every $x$-coordinate there is below zero.

The same value also comes from a right triangle. Drop the $150^\circ$ point straight down to the $x$-axis and you form a $30^\circ$ reference triangle with hypotenuse $1$, whose horizontal leg has length $\cos 30^\circ = \frac{\sqrt{3}}{2}$. Because that leg points in the negative $x$-direction, the coordinate reads $-\frac{\sqrt{3}}{2}$, so triangle and circle give the identical number, the double anchor every trig value should have. A version with the tangent line drawn in lives at unit circle with tangent.

What Are The Related Values Around Cos 150 Degrees?

The angles near $150^\circ$ share the same reference triangles, so their values rhyme. Reading them side by side makes the sign pattern of the second quadrant obvious.

Table 1: Cosine, sine, and tangent for angles near 150°, in degrees and radians.

Notice that $\cos 150^\circ$ and $\cos 30^\circ$ are equal in size and opposite in sign, and that cosine slides from $-\frac{\sqrt{3}}{2}$ down to $-1$ as the angle grows from $150^\circ$ to $180^\circ$. Each neighbour has its own reference page: $\cos 120^\circ$ and $\cos 180^\circ$ extend the pattern in one direction. For the complete grid of standard angles, the trigonometric table lists them all, and the radian-form values are collected under trigonometric ratios in radians.

Why Is Cos 150 Degrees Negative?

The sign is not a rule to memorise, it is a direct reading of position on the circle. Here is the reasoning in plain steps.

• Cosine is a horizontal coordinate. On the unit circle, cosine measures how far right or left the point sits, so its sign follows the horizontal direction.

• Past 90°, the point moves left. Any angle between $90^\circ$ and $180^\circ$ has its point in the second quadrant, to the left of the vertical axis, where the horizontal coordinate is negative.

• The size stays tied to 30°. The reference angle of $150^\circ$ is $30^\circ$, so the magnitude is $\cos 30^\circ = \frac{\sqrt{3}}{2}$, and only the direction, and therefore the sign, changes.

Put together, an angle that has opened past straight up must have a leftward, negative cosine, and its exact size is the familiar $\frac{\sqrt{3}}{2}$. That is why $\cos 150^\circ = -\frac{\sqrt{3}}{2}$ rather than the positive version.

Who Discovered The Cosine Values Behind Cos 150 Degrees?

Long before the word "cosine" existed, astronomers needed the length of a chord across a circle to track the stars, and that need built the first trigonometric tables. The story of a value like $\cos 150^\circ$ starts with people mapping the sky.

Two later figures turned those chords into the sine-based trigonometry we use today.

• Claudius Ptolemy (c. 100–170 CE, Roman Egypt) refined the chord table in his Almagest, computing chords accurate enough to serve astronomers for over a thousand years.

• Aryabhata (c. 476–550 CE, India) replaced chords with the half-chord, the jya, which became our sine, and tabulated these values in the Aryabhatiya, giving trigonometry the function that makes $\cos 150^\circ$ easy to state.

Where Is Cos 150 Degrees Used In The Real World?

A negative cosine is not a classroom curiosity, it shows up wherever something points backward, oscillates, or turns past a right angle.

• Alternating current: household electricity follows a cosine wave, and for part of every cycle the voltage is negative, exactly the regime that angles like $150^\circ$ describe.

• Projectile motion: the horizontal reach of a thrown object depends on the cosine of its launch direction, and cosines of obtuse angles model motion aimed backward or up-and-over.

• GPS and navigation: position fixes rely on phase differences between signals, and cosines of angles across all four quadrants keep the geometry consistent.

• Computer graphics: rotating a game character past $90^\circ$ feeds obtuse angles into the cosine, and the negative output is what turns the object to face left.

• Astronomy: the same chord-and-angle reasoning Hipparchus used still underlies how we compute apparent positions of planets.

One negative number, $-\frac{\sqrt{3}}{2}$, quietly runs through power grids, ballistics, satellites, and screens. The math a student meets on paper is the math those systems run on.

What Are The Most Common Mistakes With Cos 150 Degrees?

These four errors account for most lost marks on this value, and each is surfaced repeatedly in "cos 150 negative or positive" questions across public help forums.

Dropping the negative sign.

Where it slips in:

A student finds the reference angle $30^\circ$, writes $\cos 30^\circ = \frac{\sqrt{3}}{2}$, and stops there, reporting a positive answer.

Don't do this:

Do not quote the reference-angle cosine as the final value. The reference angle gives the size, never the sign.

The correct way:

Check the quadrant first. $150^\circ$ is in the second quadrant where cosine is negative, so the answer is $-\frac{\sqrt{3}}{2}$, not $+\frac{\sqrt{3}}{2}$.

Using 150° as its own reference angle.

Where it slips in:

A student tries to read a value straight off $150^\circ$ or subtracts from the wrong axis, getting a reference angle of $60^\circ$ or $150^\circ$.

Don't do this:

Do not measure the reference angle from the vertical axis or from $90^\circ$. For a second-quadrant angle it is measured from $180^\circ$.

The correct way:

Compute $180^\circ - 150^\circ = 30^\circ$. The reference angle is $30^\circ$, which is why the magnitude is $\frac{\sqrt{3}}{2}$.

Leaving the calculator in radian mode.

Where it slips in:

A student types $\cos(150)$ expecting degrees, but the calculator is set to radians and returns about $0.699$, a completely different number.

Don't do this:

Do not trust a decimal without checking the angle mode. $\cos(150 \text{ rad})$ is not $\cos 150^\circ$.

The correct way:

Set the mode to degrees before entering $150$, or convert first: $150^\circ = \frac{5\pi}{6}$ radians, then evaluate $\cos\frac{5\pi}{6} \approx -0.8660$.

Confusing cos 150° with sin 150°.

Where it slips in:

A student swaps the two, reporting $\cos 150^\circ = \frac{1}{2}$, which is actually the sine.

Don't do this:

Do not assume the second-quadrant sign is the same for both functions. Here sine is positive and cosine is negative.

The correct way:

Read the coordinates in order: the point is $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, so cosine (the $x$-value) is $-\frac{\sqrt{3}}{2}$ and sine (the $y$-value) is $\frac{1}{2}$.

Practice Problems On Cos 150 Degrees

Work each one, then check the answer that follows.

1. Convert $150^\circ$ to radians.
   (Answer: $150^\circ \times \frac{\pi}{180^\circ} = \frac{5\pi}{6}$.)

2. State the reference angle of $150^\circ$ and the quadrant.
   (Answer: reference angle $30^\circ$, second quadrant.)

3. Evaluate $\cos 150^\circ + \cos 30^\circ$.
   (Answer: $-\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = 0$.)

4. Find $\tan 150^\circ$ using the values in Table 1.
   (Answer: $\frac{\sin 150^\circ}{\cos 150^\circ} = \frac{1/2}{-\sqrt{3}/2} = -\frac{1}{\sqrt{3}} \approx -0.5774$.)

5. Use the angle-sum formula to evaluate $\cos(180^\circ - 30^\circ)$.
   (Answer: $-\cos 30^\circ = -\frac{\sqrt{3}}{2}$.)

6. Is $\cos 210^\circ$ equal to $\cos 150^\circ$?
   (Answer: yes, both equal $-\frac{\sqrt{3}}{2}$, since $210^\circ$ also has reference angle $30^\circ$ with cosine negative in the third quadrant.)

Where Should You Go Next After Cos 150 Degrees?

This value opens onto the wider structure of trigonometry, and a few natural doors lead outward.

1. Trigonometric ratios of specific angles. Learn the full set of $30^\circ$, $45^\circ$, and $60^\circ$ values that every second-quadrant answer is built from.

2. Trigonometric ratios. Step back to how sine, cosine, and tangent are defined from a right triangle before the unit circle extends them.

3. Cos 120 degrees. Apply the same reference-angle method to the neighbouring second-quadrant angle and compare the two.

If your child is learning to reason about signs and reference angles rather than memorise a table, a live Bhanzu trainer teaches trigonometry starting from the unit circle in the Bhanzu trigonometry program.