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

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, 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?

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°.