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

[Trigonometry](/content/tag/trigonometry/index.html)

TL;DR

The value of cos 4 degrees is approximately 0.9976 — it is not a special-angle exact value, so there is no clean surd for it. This article shows how to find cos⁡4° honestly (calculator and the small-angle approximation, with its accuracy bound), gives the radian form, and places it on the unit circle.

Last updated on July 15, 20226 min read

## What Does Cos 4 Degrees Mean?

Cosine of an angle on the unit circle (radius 1, centered at the origin) is the x-coordinate of the point at that angle, where every point is (cos⁡θ,sin⁡θ). 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…≈0.9976

In radian mode the same keystrokes give cos⁡(4 rad)≈−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⁡θ≈1−θ²/2 — this comes from the first terms of the cosine Taylor series. Convert first: 4°=π/45≈0.069813 rad.

cos⁡4°≈1−(0.069813)²/2=0.997563

That matches the calculator to roughly six decimal places — the error here is only about 1×10−6.

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

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

| Angle | 1−θ²/2 | True cosine | Error |
| --- | --- | --- | --- |
| 4° | 0.997563 | 0.997564 | ≈0.000001 |
| 10° | 0.984769 | 0.984808 | ≈0.00004 |
| 15° | 0.965734 | 0.965926 | ≈0.0002 |
| 20° | 0.939076 | 0.939693 | ≈0.0006 |

The rule of thumb: trust cos⁡θ≈1−θ²/2 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 π/45 rad, but the _value_ of the cosine is the same number, ≈0.9976. Converting the angle to radians does not change the cosine; it only changes how the angle is labeled.

## 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°≈1−(4²/2) = −7.

_Why it breaks._ The formula cos⁡θ≈1−θ²/2 needs θ in **radians**, not degrees.

_Correct._ Convert first: 4°=0.069813 rad, then 1−(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… 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−θ²/2.

**Don't do this:** writing cos⁡4°≈1−(4²/2) = −7.

**The correct way:** convert to radians first, then apply the formula.

### Mistake 2: Hunting for an exact surd

**Where it slips in:** assuming every angle has a clean value like cos⁡30°=√3/2.

**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.

### Mistake 3: Trusting the approximation past its range

**Where it slips in:** carrying 1−θ²/2 up to large angles because it worked at 4°.

**Don't do this:** using it for cos⁡40° and reporting it as accurate.

**The correct way:** the approximation is reliable below about 15°; past 20° its error grows past 0.0005.

## 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⁡θ≈1−θ²/2 (in radians) is accurate to four decimals below about 15° and unreliable past 20°.
- In radians the angle is π/45, but the cosine value is unchanged at ≈0.9976.
- The biggest slip is using the small-angle formula in degrees instead of radians.