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# Cos 1 Degree — Value of cos(1°) and How to Find It

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

TL;DR

The value of cos 1 degree is approximately 0.9998 — it is not a special-angle exact value, so there is no clean surd for it. This article shows how to find cos⁡1° honestly (calculator and small-angle approximation), gives the decimal and radian form, and places it among the standard angles.

BT

[Bhanzu Team](/content/authors/bhanzu-team/index.html) Last updated on June 13, 2026 6 min read

> ### **Quick Answer:**
>
> - **Result:** cos⁡1°≈0.9998
>
> - **In radians:** cos⁡!(π/180)=cos⁡(0.01745)≈0.9998
>
> - **Notation:** decimal approximation — 0.99984769 (8 dp)
>
> - **Method shown:** calculator (degree mode) and the small-angle approximation cos⁡θ≈1−θ²/2
>
> - **Exact form:** none simple — 1° is not a special angle, so no clean radical exists

## Quick Reference — Cosine Near The Small Angles

Cos 1° is sandwiched between cos⁡0°=1 and the next values. The nearest special angle is cos⁡0°.

| Angle (degrees) | Angle (radians) | cos⁡θ | Special angle? |
| --- | --- | --- | --- |
| 0° | 0 | 1.0000 | Yes (exact 1) |
| **1°** | **π/180** | **0.9998** | No — decimal only |
| 2° | π/90 | 0.9994 | No |
| 5° | π/36 | 0.9962 | No |
| 10° | π/18 | 0.9848 | No |
| 30° | π/6 | 0.8660 | Yes (√3/2) |
| 45° | π/4 | 0.7071 | Yes (√2/2) |

The closest exact landmark to cos⁡1° is cos⁡0°=1, and cos⁡1° falls only 0.0002 short of it.

## Where cos 1 Degree Shows Up

A 1-degree angle is the kind of small misalignment that matters in precision work. A telescope or antenna pointed 1° off-target still has cos⁡1°≈0.9998 of its aim on-axis — almost all of it — which is why small angular errors barely reduce signal strength but a 1-degree slope over a long horizontal run still produces a measurable rise. Surveyors and machinists treat cos⁡1° as "effectively 1" for exactly this reason.

## What Does cos 1 Degree 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).

At 1° the radius has barely turned off the positive x-axis, so the point is almost at (1,0) — its x-coordinate is about 0.9998. That is cos⁡1°.

## How Do You Find The Value of cos 1 Degree?

Because 1° is not a special angle, there is no surd to simplify to. Here are the honest routes to the value.

### **Method 1: Calculator (set to degree mode)**

Type cos⁡(1) with the calculator in **DEG** mode.

cos⁡1°=0.99984769...≈0.9998

If the calculator is in radian mode you would get cos⁡(1 rad)≈0.5403 — a completely different number, so the mode matters.

### **Method 2: Small-angle approximation**

For small angles in radians, cos⁡θ≈1−θ²/2. Convert first: 1°=π/180≈0.017453 rad.

cos⁡1°≈1−(0.017453)²/2=1−0.000152=0.999848

That matches the calculator to four decimal places — the approximation is accurate because 1° is tiny.

## Examples Using cos 1 Degree

### Example 1

**State cos⁡1° to four decimal places.**

From a calculator in degree mode, cos⁡1°=0.9998.

### Example 2 (wrong path first)

**Find cos⁡1° using the small-angle formula.**

_Wrong attempt._ A student plugs the degree value straight in: cos⁡1°≈1−1²/2=0.5.

_Why it breaks._ The formula cos⁡θ≈1−θ²/2 needs θ in **radians**, not degrees. Using 1 (degrees) treats the angle as 1 radian — about 57° — which is why the answer collapsed to 0.5.

_Correct._ Convert first: 1°=0.01745 rad, then 1−(0.01745)²/2=0.9998.

### Example 3

**A radio dish is aimed 1° off a satellite. What fraction of its pointing is on-axis?**

The on-axis fraction is cos⁡1°=0.9998, so 99.98% of the aim is on-target.

### Example 4

**Compare cos⁡1° with cos⁡0°.**

cos⁡0°=1 exactly; cos⁡1°=0.9998. The gap is just 0.0002 — a single degree barely dents cosine near the top.

### Example 5

**Round cos⁡1° to two decimal places.**

0.99984 rounds to 1. To two places, cos⁡1° is indistinguishable from cos⁡0°.

## Cos 1 Degree — 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⁡1°≈1−1²/2=0.5.

**The correct way:** convert to radians first (1°=0.01745 rad), then apply the formula.

### Mistake 2: Hunting for an exact surd

*Don't do this:* trying to write cos⁡1° as a simple radical.

**The correct way:** 1° is not a special angle, so cos⁡1° is given as the decimal 0.9998.

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

**Don't do this:** reading cos⁡(1)=0.5403 and reporting it as cos⁡1°.

**The correct way:** check **DEG** mode for cos⁡1°; 0.5403 is cos⁡(1 rad), a different angle entirely.

## The short version

- **Cos 1 degree** is approximately 0.9998 — a decimal, not a clean surd.
- 1° is not a special angle, so the value comes from a calculator or the small-angle approximation.
- In radians the angle is π/180, but the cosine value is unchanged at ≈0.9998.
- The biggest slip is using the small-angle formula in degrees instead of radians.
- cos⁡1° sits a mere 0.0002 below cos⁡0°=1.