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 $\cos 0°$.
Angle (degrees) | Angle (radians) | $\cos\theta$ | Special angle? |
|---|---|---|---|
$0°$ | $0$ | $1.0000$ | Yes (exact $1$) |
$1°$ | $\pi/180$ | $0.9998$ | No — decimal only |
$2°$ | $\pi/90$ | $0.9994$ | No |
$5°$ | $\pi/36$ | $0.9962$ | No |
$10°$ | $\pi/18$ | $0.9848$ | No |
$30°$ | $\pi/6$ | $0.8660$ | Yes ($\tfrac{\sqrt3}{2}$) |
$45°$ | $\pi/4$ | $0.7071$ | Yes ($\tfrac{\sqrt2}{2}$) |
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 $\cos 1° \approx 0.9998$ 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
State $\cos 1°$ to four decimal places.
Use $\cos\theta \approx 1 - \tfrac{\theta^2}{2}$ (in radians) to estimate $\cos 1°$.
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.
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