Cos 60 Degrees : Exact Value, 1/2, and How to Find It

What Does Cos 60 Degrees Mean?

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 $\cos 60^\circ$ asks what fraction of the hypotenuse the adjacent side is when the angle is $60^\circ$.

On the unit circle, a circle of radius $1$ centred at the origin, cosine is the $x$-coordinate of the point where the angle's radius meets the circle. At $60^\circ$ that point is $\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$, so the cosine is $\dfrac{1}{2}$.

Where Does Cos 60 Degrees Show Up?

A force pushed at $60^\circ$ to the horizontal keeps only half its size in the horizontal direction, because that component scales with $\cos 60^\circ = \dfrac{1}{2}$. A $10\text{ N}$ pull at $60^\circ$ therefore drags with just $5\text{ N}$ sideways.

The same half appears in the geometry of a regular hexagon and in any equilateral-triangle construction, where the $60^\circ$ corner sits on the unit circle at the point $\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$.

Standard-Angle Reference Table

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

Read the column top to bottom and cosine slides from $1$ down to $0$. The radian twin of this value lives at cos π/3, where the same $\dfrac{1}{2}$ is reached from the unit circle instead of the triangle.

How Do You Find The Exact Value Of Cos 60 Degrees?

Two clean routes both land on $\dfrac{1}{2}$: one builds it from a triangle, the other reads it off the unit circle.

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 $30^\circ$, $60^\circ$, and $90^\circ$.

In one of those right triangles the hypotenuse is $2$, the side adjacent to the $60^\circ$ angle is $1$ (half the base), and the remaining side is $\sqrt{3}$. Apply the definition:

$$\cos 60^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{2}$$

Method 2: The unit circle.

Set the radius to $1$ and rotate it $60^\circ$ above the positive $x$-axis. The tip lands at $\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$, and the $x$-coordinate is the cosine.

$$\cos 60^\circ = x\text{-coordinate} = \frac{1}{2}$$

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

Examples Of Cos 60 Degrees

Example 1

Evaluate $8\cos 60^\circ$.

$$8\cos 60^\circ = 8 \times \frac{1}{2} = 4$$

Example 2

A student is asked for $\cos 60^\circ$ and reasons that, since $60^\circ$ is larger than $30^\circ$, its cosine must be larger too. What goes wrong?

Wrong attempt. The student writes $\cos 60^\circ = \dfrac{\sqrt{3}}{2} \approx 0.87$, borrowing the bigger value.

That breaks against the table: cosine shrinks as the angle opens toward $90^\circ$, so the larger angle must have the smaller cosine.

Correct. $\cos 60^\circ = \dfrac{1}{2} = 0.5$, which is smaller than $\cos 30^\circ = \dfrac{\sqrt{3}}{2}$. The value $\dfrac{\sqrt{3}}{2}$ belongs to $30^\circ$, not $60^\circ$.

Example 3

A right triangle has a hypotenuse of $12\text{ cm}$ and a $60^\circ$ angle. Find the side adjacent to that angle.

$$\cos 60^\circ = \frac{\text{adjacent}}{12} \implies \text{adjacent} = 12 \times \frac{1}{2} = 6 \text{ cm}$$

Example 4

Verify $\cos^2 60^\circ + \sin^2 60^\circ = 1$.

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

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

Example 5

Use the cofunction relationship to check $\cos 60^\circ$ against a sine value.

Cosine and sine are cofunctions: $\cos\theta = \sin(90^\circ - \theta)$. So $\cos 60^\circ = \sin 30^\circ = \dfrac{1}{2}$, which matches.

Where Students Trip Up On Cos 60 Degrees

Mistake 1: Swapping cos 60 and cos 30

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

Don't do this: Writing $\cos 60^\circ = \dfrac{\sqrt{3}}{2}$. That is $\cos 30^\circ$.

The correct way: The larger angle carries the smaller cosine. The memorizer who anchors on "cosine starts at $1$ and shrinks" stops swapping the two, because $60^\circ$ is closer to $90^\circ$ and so must sit nearer $0$.

Mistake 2: Reading cos 60 as sin 60

Where it slips in: Copying a value across from a sine row on a crowded table.

Don't do this: Writing $\cos 60^\circ = \dfrac{\sqrt{3}}{2}$ because $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$.

The correct way: At $60^\circ$ the two functions cross over: $\cos 60^\circ = \dfrac{1}{2}$ and $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$. The student who never separates the sine and cosine rows is the one who keeps meeting this error on the next problem.

Mistake 3: Leaving the calculator in radian mode

Where it slips in: Entering $\cos(60)$ on a calculator still set to radians.

Don't do this: Trusting the screen reading of about $-0.952$ without checking the mode.

The correct way: Confirm degree mode before entering $\cos(60)$; a value nowhere near $0.5$ is the signal the mode is wrong.

Key Takeaways

• Cos 60 degrees equals $\dfrac{1}{2}$, exactly $0.5$, an exact value because $60^\circ$ is a standard angle.

• The 30-60-90 triangle gives it as adjacent over hypotenuse; the unit circle gives it as the $x$-coordinate at $60^\circ$.

• In radians, $\cos 60^\circ = \cos\left(\dfrac{\pi}{3}\right)$.

• The most common slip is borrowing $\dfrac{\sqrt{3}}{2}$ from $30^\circ$, since cosine shrinks as the angle grows.

To take this angle further with a teacher, explore Bhanzu's trigonometry tutor, its high school math tutor programme, or live math classes online.

Practice These Before Moving On

1. Evaluate $6\cos 60^\circ + 2\sin 30^\circ$.

2. A ramp meets the ground at $60^\circ$ with a $9\text{ m}$ slope length. Use $\cos 60^\circ$ to find its horizontal reach.

3. Show that $\cos 60^\circ \times \cos 0^\circ = \cos 60^\circ$.

Want a live Bhanzu trainer to walk through more cos 60 degrees problems? Book a free demo class.

Read More

• Cos 90 Degrees — the next standard angle, where cosine reaches $0$.

• Cos 30 Degrees — the mirror partner whose value is $\dfrac{\sqrt{3}}{2}$.

• Sin, Cos, Tan — how the three ratios are defined together.

• Trigonometric Table — every standard angle in one chart.

• Reference Angle — how angles past $90^\circ$ borrow a first-quadrant value.