Cos 30 Degrees - Value √3/2 Explained

The value of cos 30 degrees is $\frac{\sqrt{3}}{2}$, or approximately $0.8660$.

Quick Answer:

• Result: $\cos 30° = \dfrac{\sqrt{3}}{2}$

• Decimal: $\approx 0.8660$

• In radians: $\cos\left(\dfrac{\pi}{6}\right) = \dfrac{\sqrt{3}}{2}$

• Exact form: $\dfrac{\sqrt{3}}{2}$ (a standard angle — the value is exact, not a rounded decimal)

• Methods shown: 30-60-90 triangle ratio · unit circle x-coordinate

Standard-Angle Cosine Reference Table

Thirty degrees is one of a handful of angles whose cosine has a clean exact form. Here are the standard first-quadrant angles in both degrees and radians.

Read the column top to bottom and cosine slides from $1$ down to $0$ — it shrinks as the angle opens up. Cos 30° and cos 60° are mirror partners: $\cos 30° = \sin 60°$ and $\cos 60° = \sin 30°$, which is the cofunction relationship at work.

Where Cos 30 Degrees Shows Up

A 30° slope is the angle of a standard wheelchair-access ramp at its steepest permitted grade, and the horizontal reach of that ramp scales with $\cos 30°$. The same value sets the spacing of bolt holes on a hexagonal nut, where each face sits 60° apart and the wrench geometry uses $\cos 30° = \frac{\sqrt{3}}{2}$.

In physics, a projectile launched at 30° travels a horizontal distance proportional to $\cos 30°$, the same Pythagorean relationship behind any inclined surface. The exact $\frac{\sqrt{3}}{2}$ value sits on the unit circle, the standard reference for every special angle.

What Cos 30 Degrees Means

Cosine is one of the three core trigonometric ratios — in a right triangle, the cosine of an angle is the side adjacent to it divided by the hypotenuse. So $\cos 30°$ asks: in a right triangle with a 30° angle, what fraction of the hypotenuse is the adjacent side?

On the unit circle — a circle of radius $1$ centred at the origin — the cosine of an angle is the $x$-coordinate of the point where the angle's radius meets the circle. At $30°$, that point is $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, so the $x$-coordinate, and therefore the cosine, is $\frac{\sqrt{3}}{2}$.

How Do You Find the Exact Value of Cos 30 Degrees?

There are two clean routes: one builds the value from a triangle, the other reads it off the unit circle. Both give $\frac{\sqrt{3}}{2}$.

Method 1: The 30-60-90 triangle

Take an equilateral triangle with each side $2$ units and drop a perpendicular from one vertex to the opposite side. That splits it into two identical right triangles, each with angles $30°$, $60°$, and $90°$.

In one of those right triangles:

• the hypotenuse is $2$ (a full side of the equilateral triangle),

• the side opposite $30°$ is $1$ (half of the base that got split),

• the side adjacent to $30°$ is $\sqrt{3}$, from the Pythagorean theorem: $\sqrt{2^2 - 1^2} = \sqrt{3}$.

Now apply the definition:

$$\cos 30° = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}$$

Method 2: The unit circle

Set the radius to $1$ and rotate it $30°$ above the positive $x$-axis. The tip lands at $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.

$$\cos 30° = x\text{-coordinate} = \frac{\sqrt{3}}{2}$$

The two methods agree because the unit circle is just the 30-60-90 triangle scaled so the hypotenuse equals $1$.

Method 3: From the decimal (calculator check)

Set the calculator to degree mode and enter $\cos(30)$, which returns $0.8660254\ldots$. Squaring $\frac{\sqrt{3}}{2}$ gives $\frac{3}{4} = 0.75$, whose square root is the same $0.8660$ — the decimal confirms the exact form.

Examples of Cos 30 Degrees

Example 1

Evaluate $4\cos 30°$.

$$4\cos 30° = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3} \approx 3.464$$

Example 2

Find $\cos 30°$ given that $\cos 30° = \sin\theta$ for an acute angle $\theta$. What is $\theta$?

Wrong attempt. A student writes $\theta = 30°$, reasoning that if the values are equal the angles must be equal.

That breaks immediately: $\sin 30° = \frac{1}{2}$, not $\frac{\sqrt{3}}{2}$, so $\theta = 30°$ gives the wrong number.

Correct. Cosine and sine are cofunctions: $\cos\theta = \sin(90° - \theta)$, so $\cos 30° = \sin 60°$ and $\theta = 60°$. Check: $\sin 60° = \frac{\sqrt{3}}{2}$, which matches.

Example 3

A right triangle has a hypotenuse of $10$ cm and a $30°$ angle. Find the length of the side adjacent to the $30°$ angle.

$$\cos 30° = \frac{\text{adjacent}}{10} \implies \text{adjacent} = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66 \text{ cm}$$

Example 4

Verify the identity $\cos^2 30° + \sin^2 30° = 1$.

$$\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = \frac{3}{4} + \frac{1}{4} = 1$$

The Pythagorean identity holds, as it must for every angle.

Example 5

Express $\cos 30°$ in radians and evaluate $\cos\left(\frac{\pi}{6}\right)$.

Since $30° = \frac{\pi}{6}$ radians, $\cos\left(\frac{\pi}{6}\right) = \cos 30° = \frac{\sqrt{3}}{2}$. The radian form and the degree form name the same angle and the same value.

Where Students Trip Up on Cos 30 Degrees

Mistake 1: Swapping cos 30° and cos 60°

Where it slips in: Recall under time pressure, when the $\frac{1}{2}$ and the $\frac{\sqrt{3}}{2}$ get attached to the wrong angle.

Don't do this: Writing $\cos 30° = \frac{1}{2}$. That is $\cos 60°$, not $\cos 30°$.

The correct way: The larger angle has the smaller cosine. $\cos 30° = \frac{\sqrt{3}}{2} \approx 0.87$ (close to $1$, because $30°$ is close to $0°$); $\cos 60° = \frac{1}{2}$.

Mistake 2: Leaving the answer as a rounded decimal when an exact value is asked

Where it slips in: Calculator-first solving, where the screen reads $0.866$ and the student copies that.

Don't do this: Writing $\cos 30° = 0.866$ on a problem that asks for the exact value.

The correct way: For a standard angle, give the exact radical form $\frac{\sqrt{3}}{2}$. The decimal $0.8660$ is an approximation; $\frac{\sqrt{3}}{2}$ is the value.

Mistake 3: Forgetting the calculator's angle mode

Where it slips in: A calculator left in radian mode returns $\cos(30) \approx 0.1543$ instead of $0.8660$.

Don't do this: Trusting the screen without checking whether it is set to degrees.

The correct way: Confirm degree mode before entering $\cos(30)$; the memorizer who never checks the mode is the one most surprised by a wrong answer here.

Key Takeaways

• Cos 30 degrees equals $\frac{\sqrt{3}}{2}$, approximately $0.8660$ — an exact value because $30°$ is a standard angle.

• The 30-60-90 triangle gives it as adjacent over hypotenuse, $\frac{\sqrt{3}}{2}$; the unit circle gives it as the $x$-coordinate at $30°$.

• In radians, $\cos 30° = \cos\left(\frac{\pi}{6}\right)$.

• The most common slip is confusing it with $\cos 60° = \frac{1}{2}$ — remember cosine shrinks as the angle grows.

Practice These Before Moving On

1. Evaluate $2\cos 30° - \tan 30°$.

2. A ramp rises at $30°$ over a horizontal run of $4$ m. Use $\cos 30°$ to find the ramp's length along the slope.

3. Show that $\cos 30° \cdot \cos 60° + \sin 30° \cdot \sin 60° = \cos 30°$.

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