Cos 4 Degrees — Value of cos(4°) and How to Find It

What Does Cos 4 Degrees Mean?

Cosine of an angle on the unit circle (radius $1$, centred at the origin) is the $x$-coordinate of the point at that angle, where every point is $(\cos\theta, \sin\theta)$. A quadrant is one of the four regions the axes cut the plane into, numbered anticlockwise from the top right; $4°$ lands in Quadrant I, where cosine is positive.

At $4°$ the radius has barely turned off the positive $x$-axis, so the point is almost at $(1, 0)$ — its $x$-coordinate is about $0.9976$. That is $\cos 4°$.

How Do You Find the Value of Cos 4 Degrees?

Because $4°$ is not a special angle, there is no surd to simplify to. Here are the two honest routes — and for a tiny angle like this, the approximation is genuinely useful.

Method 1: Calculator (set to degree mode)

Type $\cos(4)$ with the calculator in DEG mode.

$$\cos 4° = 0.99756405\ldots \approx 0.9976$$

In radian mode the same keystrokes give $\cos(4\ \text{rad}) \approx -0.6536$ — a completely different number, so the mode matters.

Method 2: Small-angle approximation (with its validity bound)

For small angles measured in radians, $\cos\theta \approx 1 - \dfrac{\theta^2}{2}$ — this comes from the first terms of the cosine Taylor series. Convert first: $4° = \dfrac{\pi}{45} \approx 0.069813$ rad.

$$\cos 4° \approx 1 - \frac{(0.069813)^2}{2} = 1 - 0.002437 = 0.997563$$

That matches the calculator to roughly six decimal places — the error here is only about $1 \times 10^{-6}$.

How accurate is the small-angle approximation, and when does it break?

It is excellent for tiny angles and degrades as the angle grows:

The rule of thumb: trust $\cos\theta \approx 1 - \tfrac{\theta^2}{2}$ to four decimal places below about $10°$–$15°$, and stop relying on it past roughly $20°$, where the error grows past $0.0005$. At $4°$ you are deep in the safe zone.

What is cos 4 degrees in radians?

The angle converts to $\frac{\pi}{45}$ rad, but the value of the cosine is the same number, $\approx 0.9976$. Converting the angle to radians does not change the cosine; it only changes how the angle is labelled.

Examples Using Cos 4 Degrees

Example 1

State $\cos 4°$ to four decimal places.

From a calculator in degree mode, $\cos 4° = 0.9976$.

Example 2 (wrong path first)

Find $\cos 4°$ using the small-angle formula.

Wrong attempt. A student plugs the degree value straight in: $\cos 4° \approx 1 - \dfrac{4^2}{2} = 1 - 8 = -7$.

Why it breaks. The formula $\cos\theta \approx 1 - \tfrac{\theta^2}{2}$ needs $\theta$ in radians, not degrees. Using $4$ (degrees) treats the angle as $4$ radians — about $229°$ — which is why the answer became a nonsensical $-7$ (cosine can never leave $[-1, 1]$).

Correct. Convert first: $4° = 0.069813$ rad, then $1 - \dfrac{(0.069813)^2}{2} = 0.9976$.

Example 3

A laser is aimed $4°$ off a distant sensor. What fraction of its pointing is on-axis?

The on-axis fraction is $\cos 4° = 0.9976$, so $99.76$ of the aim is on-target.

Example 4

Compare $\cos 4°$ with $\cos 0°$.

$\cos 0° = 1$ exactly; $\cos 4° = 0.9976$. The gap is just $0.0024$ — a few degrees barely dent cosine near the top.

Example 5

Round $\cos 4°$ to two decimal places.

$0.99756\ldots$ rounds to $1.00$. To two places, $\cos 4°$ is indistinguishable from $\cos 0°$.

Cos 4 Degrees — Where Things Go Sideways

Most errors on a small non-special angle come from the same few habits.

Mistake 1: Using the small-angle formula in degrees

Where it slips in: plugging the raw degree number into $1 - \tfrac{\theta^2}{2}$.

Don't do this: writing $\cos 4° \approx 1 - \dfrac{4^2}{2} = -7$.

The correct way: convert to radians first ($4° = 0.069813$ rad), then apply the formula. The habit that fixes this is to circle the word "radians" in the formula before any number goes in; the learner who skips the conversion treats the angle as radians and lands wildly off, sometimes outside cosine's own range.

Mistake 2: Hunting for an exact surd

Where it slips in: assuming every angle has a clean value like $\cos 30° = \tfrac{\sqrt3}{2}$.

Don't do this: trying to write $\cos 4°$ as a simple radical.

The correct way: $4°$ is not a special angle, so $\cos 4°$ is given as the decimal $0.9976$. The learner who only knows the special-angle table has to switch to a calculator or the small-angle approximation here — and that is the honest answer, not a failure.

Mistake 3: Trusting the approximation past its range

Where it slips in: carrying $1 - \tfrac{\theta^2}{2}$ up to large angles because it worked at $4°$.

Don't do this: using it for $\cos 40°$ and reporting $1 - \tfrac{(0.698)^2}{2} = 0.756$ as accurate.

The correct way: the approximation is reliable below about $15°$; past $20°$ its error grows past $0.0005$, and at $40°$ it is off by more than $0.01$. Beyond the small-angle zone, use the calculator.

Key Takeaways

• Cos 4 degrees is approximately $0.9976$ — a decimal, not a clean surd.

• $4°$ is not a special angle, so the value comes from a calculator or the small-angle approximation.

• The approximation $\cos\theta \approx 1 - \tfrac{\theta^2}{2}$ (in radians) is accurate to four decimals below about $15°$ and unreliable past $20°$.

• In radians the angle is $\frac{\pi}{45}$, but the cosine value is unchanged at $\approx 0.9976$.

• The biggest slip is using the small-angle formula in degrees instead of radians.

To take cosine values further with a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or math tutoring.

Practice These Before Moving On

1. State $\cos 4°$ to four decimal places.

2. Use $\cos\theta \approx 1 - \tfrac{\theta^2}{2}$ (in radians) to estimate $\cos 4°$.

3. Explain why the small-angle approximation is trustworthy at $4°$ but not at $40°$.

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Read More

• Cos 2 Degrees — value and how to find it

• Cos 10 Degrees — value and how to find it

• Cos 0 Degrees — the exact landmark value

• Trigonometric Functions and their graphs

• Trigonometric Table — ratios for standard angles