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

Quick Answer:

• Result: $\cos 1° \approx 0.9998$

• In radians: $\cos!\left(\frac{\pi}{180}\right) = \cos(0.01745) \approx 0.9998$

• Notation: decimal approximation — $0.99984769$ (8 dp)

• Method shown: calculator (degree mode) and the small-angle approximation $\cos\theta \approx 1 - \tfrac{\theta^2}{2}$

• 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 $\cos 0°$.

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 $\cos 1° \approx 0.9998$ 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 $1$, centred at the origin) is the $x$-coordinate of the point at that angle, where every point is $(\cos\theta, \sin\theta)$.

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.

$$\cos 1° = 0.99984769\ldots \approx 0.9998$$

If the calculator is in radian mode you would get $\cos(1\ \text{rad}) \approx 0.5403$ — a completely different number, so the mode matters.

Method 2: Small-angle approximation

For small angles in radians, $\cos\theta \approx 1 - \dfrac{\theta^2}{2}$. Convert first: $1° = \dfrac{\pi}{180} \approx 0.017453$ rad.

$$\cos 1° \approx 1 - \frac{(0.017453)^2}{2} = 1 - 0.000152 = 0.999848$$

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

What is cos 1 degree in radians? The angle converts to $\frac{\pi}{180}$ rad, but the value of the cosine is the same number, $\approx 0.9998$. Converting the angle to radians does not change the cosine; it only changes how the angle is labelled.

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: $\cos 1° \approx 1 - \frac{1^2}{2} = 0.5$.

Why it breaks. The formula $\cos\theta \approx 1 - \tfrac{\theta^2}{2}$ needs $\theta$ 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 $1 - \frac{(0.01745)^2}{2} = 0.9998$.

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.

$0.99984\ldots$ rounds to $1.00$. 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 $1 - \tfrac{\theta^2}{2}$.

Don't do this: writing $\cos 1° \approx 1 - \frac{1^2}{2} = 0.5$.

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

Mistake 2: Hunting for an exact surd

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

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

The correct way: $1°$ is not a special angle, so $\cos 1°$ is given as the decimal $0.9998$. The memoriser who only knows the special-angle table has to switch to a calculator or approximation here — and that is the honest answer, not a failure.

Mistake 3: Forgetting the calculator's angle mode

Where it slips in: the calculator was left in radian 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 $\cos(1\ \text{radian})$, 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 $\frac{\pi}{180}$, but the cosine value is unchanged at $\approx 0.9998$.

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

• $\cos 1°$ sits a mere $0.0002$ below $\cos 0° = 1$.

Try these three before moving on

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

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

3. Explain why $\cos 1°$ is almost equal to $\cos 0°$.

If #2 gave you $0.5$, you used degrees in a radians formula — convert first. Want a live Bhanzu trainer to walk through more cosine-value problems? Book a free demo class — online globally.