# Cos 30 Degrees - Value √3/2 Explained

## TL;DR

The value of cos 30 degrees is exactly \( \frac{\sqrt{3}}{2} \), which is about 0.866. This article shows where that value comes from using the 30-60-90 triangle and the unit circle, gives a standard-angle reference table in both degrees and radians, and walks through worked examples and the mistakes students make.

**The value of** **cos 30 degrees** **is** \( \frac{\sqrt{3}}{2} \), **or approximately** 0.866.

### Quick Answer:

- **Result:** \( cos 30° = \frac{\sqrt{3}}{2} \)
- **Decimal:** ≈ 0.866
- **In radians:** \( cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \)
- **Exact form:** \( \frac{\sqrt{3}}{2} \) (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

| Angle (degrees) | Angle (radians)    | cos⁡θ (exact)   | cos⁡θ (decimal)  |
|------------------|---------------------|------------------|-------------------|
| 0°               | 0                   | 1                | 1.0000            |
| 30°              | \( \frac{\pi}{6} \)  | \( \frac{\sqrt{3}}{2} \)   | 0.8660            |
| 45°              | \( \frac{\pi}{4} \)  | \( \frac{\sqrt{2}}{2} \)    | 0.7071            |
| 60°              | \( \frac{\pi}{3} \)  | \( \frac{1}{2} \)           | 0.5000            |
| 90°              | \( \frac{\pi}{2} \)  | 0                | 0.0000            |

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 \( cos 60° = sin 30° \).

## Where Cos 30 Degrees Shows Up

A 30° slope is the angle of a standard wheelchair-access ramp at its steepest permitted grade. The same value sets the spacing of bolt holes on a hexagonal nut, where each face sits 60° apart and 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 value sits on the [unit circle](/content/math/geometry/unit-circle/index.html), the standard reference for every special angle.

## What Cos 30 Degrees Means

Cosine is one of the three core [trigonometric ratios](/content/math/trigonometry/trigonometric-ratios/index.html) — 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?

### 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,
- the side opposite 30° is 1,
- the side adjacent to 30° is \( \sqrt{3} \), from the Pythagorean theorem: \( \sqrt{2^2 - 1^2} = \sqrt{3} \).

Now apply the definition:
\[ cos 30° = \frac{adjacent}{hypotenuse} = \frac{\sqrt{3}}{2} \]

### Method 2: The unit circle

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

\[ cos 30° = x-coordinate = \frac{\sqrt{3}}{2} \]

### Method 3: From the decimal (calculator check)

Set the calculator to degree mode and enter \( cos(30) \), which returns approximately 0.866. Squaring \( \frac{\sqrt{3}}{2} \) gives 0.75, whose square root is the same 0.866 — 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 \( cos 30° = sin \theta \). What is \( \theta \)?**

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

Correct: Cosine and sine are cofunctions: \( cos \theta = sin(90° - \theta) \), so \( \theta = 60° \).

### 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.**
\[ cos 30° = \frac{adjacent}{10} \implies adjacent = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66 \text{ cm} \]

### Example 4
**Verify the identity \( cos^2 30° + sin^2 30° = 1 \).**
\[ \left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = \frac{3}{4} + \frac{1}{4} = 1 \]

### Example 5
**Express \( cos 30° \) in radians and evaluate \( cos(\frac{\pi}{6}) \).**

Since 30° equals \( \frac{\pi}{6} \) radians, we have \( cos(\frac{\pi}{6}) = cos 30° = \frac{\sqrt{3}}{2} \).

## Where Students Trip Up on Cos 30 Degrees

### Mistake 1: Swapping cos 30° and cos 60°

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

### Mistake 3: Forgetting the calculator's angle mode

## Key Takeaways

- Cos 30 degrees equals \( \frac{\sqrt{3}}{2} \), approximately 0.866 — an exact value.
- 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°.