Book A Free Math Class

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

Trigonometry

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:

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.