Cosine Error in Police Radar and Lidar Explained

By I Go Cars · Updated October 29, 2026

What cosine error is

Cosine error, which NHTSA calls the cosine effect, is the difference between how fast a vehicle is actually traveling and the figure a radar or lidar device measures when the device is not directly in the vehicle's path. The device sees only the part of the vehicle's motion that is straight toward or away from it, so the displayed reading equals the true value multiplied by the cosine of the angle between the vehicle's direction of travel and the device's line of sight.

The National Highway Traffic Safety Administration (NHTSA) defines it in its Down-the-Road Radar Module specification (DOT HS 812 266), the federal performance standard for police radar. Section 1.4.6 relates the effect to "the cosine of the angle between the direction of travel and a line from the radar device to the target." The specification judges a radar's accuracy "after accounting for cosine angle effect" (§1.4.12). The cosine is a trigonometry value: 1.0 at zero degrees, falling toward zero as the angle approaches 90 degrees.

For a parked radar or a lidar gun, the effect works in one direction only. It lowers the reading. NHTSA's 2018 radar instructor manual puts it plainly: the cosine effect "will always be to the motorist's advantage."

Stationary radar and lidar: the reading can only go down

NHTSA's radar training manual explains why. "R.A.D.A.R. will only measure the exact speed of a target vehicle when it is traveling directly at or straight from" the device. At any other angle, "The cosine effect only results in a stationary R.A.D.A.R. device displaying a speed less than the target vehicle's" true speed. At 90 degrees, with the car crossing directly in front of the antenna, the radar sees no closing motion at all.

The specification says the same thing in engineering terms: the cosine effect always lowers the measured Doppler shift (the frequency change from which radar computes velocity). That a lower reading favors the motorist is an interpretation, but it is the one NHTSA's own training manual states outright.

Lidar behaves the same way. NHTSA's Lidar Participant Manual says, "This effect always works to the motorist's advantage," and that it "does not become a factor until the angle reaches about 10 degrees."

How big the error is: the math

The displayed figure equals the true figure multiplied by the cosine of the angle. For a car traveling at a true 60 mph:

1 degree: 60.0 mph displayed.
5 degrees: 59.8 mph.
10 degrees: 59.1 mph (cosine 0.9848, about 1.5% low).
15 degrees: 58.0 mph.
20 degrees: 56.4 mph.
30 degrees: 52.0 mph (cosine 0.866).
45 degrees: 42.4 mph.
60 degrees: 30.0 mph.
90 degrees: 0 mph.

This is why NHTSA says "The cosine effect is not significant if the angle remains small. It starts becoming a factor at about 10 degrees." Below 10 degrees, the error stays under about 1.5%. The belief that a few degrees of angle changes a ticket is not supported by the arithmetic.

A realistic roadside setup

NHTSA gives operators a rule of thumb: "10 feet from the edge of the roadway for every 100 feet the target vehicle is down the roadway," and "Align the antenna as straight down the road as possible." A radar 50 feet from the travel lane, reading a car 500 feet away, sits at an angle of about 5.7 degrees (the arctangent of 50/500). The reading is about 99.5% of the true value. A car at a true 70 mph would display about 69.7 mph.

Moving radar: usually low, sometimes high

Moving radar is the one case where cosine error can work against the driver. In moving mode the radar calculates the target's velocity from two measurements: how fast the patrol car itself is going, read from the road or roadside, and the closing rate between the cruiser and the target. An angle can affect either.

NHTSA's manual states the balance precisely: "The cosine effect with moving R.A.D.A.R. may result in a high target vehicle speed display, but most often, the cosine effect with moving R.A.D.A.R. will result in a low target vehicle's speed, just as in stationary operations." The condition that decides which way it goes is also stated: "A low target speed reading can result only if the patrol vehicle speed is being computed correctly. The computation of a higher target speed could result if a less-than-true patrol speed is used by the R.A.D.A.R. device."

Worked example: the normal case, reading low

A patrol car at 50 mph meets an oncoming car at 60 mph. The two are closing at 110 mph. If the closing signal arrives at 10 degrees, the radar measures 110 × 0.9848 = 108.3 mph. Subtracting the cruiser's correctly measured 50 mph gives 58.3 mph. The display reads 1.7 mph low.

Worked example: the exception, reading high

Now suppose the radar takes the cruiser's own movement off a roadside fence at a 20-degree angle. NHTSA describes exactly this: an improper target speed results "if the patrol speed signal is received from some roadside object at an angle ... e.g., a fence by the side of the road." The radar computes the cruiser's pace as 50 × 0.9397 = 47.0 mph. The target is then computed as 110 − 47.0 = 63.0 mph, 3.0 mph above the truth. "The target speed will be high because the R.A.D.A.R.'s reading of the patrol vehicle's speed is low."

Shadowing: the related moving-mode error

Shadowing produces the same high-reading result by a different route. NHTSA's SMD Glossary, the defined-terms list for its Speed-Measuring Device Operator Training, describes it as "A patrol speed display (lower than actual) which occurs when a moving mode R.A.D.A.R. uses a slow moving vehicle" "traveling in the same direction as the patrol vehicle" in place of the road as its reference. The patrol figure reads low; the target figure reads high. The glossary also notes that in moving mode "cosine may effect the patrol speed," which ties the two errors together.

The specification guards the patrol channel separately: §2.12.8.2 of DOT HS 812 266 requires the patrol signal channel to track the cruiser within ±1 mph while it accelerates or slows. The overall moving-mode tolerance for patrol and target readings is ±2 mph.

How an officer catches it

Both high-reading cases share one fingerprint: the displayed patrol figure is lower than how fast the cruiser is actually going. NHTSA says the fence condition "normally occurs only for a short period and is easily detected by the operator" by comparing the displayed patrol figure with the speedometer. In the fence example, the radar shows 47 mph while the speedometer reads 50. Shadowing is caught the same way, by checking that figure as part of the tracking history, the sequence of visual, audio and device confirmations NHTSA trains officers to follow.

For anyone reviewing a moving-radar reading, that is the relevant question: did the officer check the displayed patrol figure against the speedometer? The NHTSA radar instructor manual includes "Verify Patrol Speed" in its opposite-lane tracking history. Whether courts in a given state accept a cosine or shadowing argument is a matter of state law and was not researched for this page; this is not legal advice.

Speed cameras and cosine compensation

Some fixed, unmanned radar systems, the kind used in automated traffic enforcement, are mounted at a set angle to the road. The NHTSA down-the-road radar specification excludes from its scope unmanned fixed-angle radars that "use circuitry to compensate for the reduction in displayed speed due to the cosine effect" (§1.3). Those devices correct for the angle electronically, so the stationary rule that the error always favors the driver describes handheld and patrol-car radar, not that class of fixed equipment.

Three misconceptions, corrected

"Cosine error can get you a higher ticket." False for stationary radar and lidar. True only for moving radar when patrol speed is under-read.
"Moving radar always errs high." NHTSA says moving-mode cosine "most often" errs low.
"A few degrees of angle matters." Under 10 degrees the error is under about 1.5%, and it is in the driver's favor.

Frequently asked questions

What is cosine error in police radar?

Cosine error is the gap between a vehicle's true velocity and the radar reading when the radar is at an angle to the vehicle's path. The radar sees only the motion directly toward or away from it, so the display equals the true value times the cosine of the angle. NHTSA's radar specification defines it in section 1.4.6.

Does cosine error favor the driver?

For stationary radar and lidar, always. NHTSA's training manuals say the cosine effect works to the motorist's advantage because it can only lower the reading. In moving radar it usually reads low as well, but it can read high when the radar under-reads how fast the patrol car itself is moving, for example from a fence or a slower same-direction vehicle.

How big is cosine error at 10 degrees?

About 1.5 percent. At 10 degrees the cosine is 0.9848, so a car traveling a true 60 mph displays about 59.1 mph. NHTSA's radar manual says the cosine effect is not significant at small angles and starts becoming a factor at about 10 degrees; its lidar manual gives the same threshold.

Can cosine error make radar read higher than my actual speed?

Only with moving radar, and only when the cruiser's own pace is under-read. If the radar takes that figure off a roadside fence at an angle or off a slower vehicle ahead, the patrol figure reads low and the computed target figure reads high. NHTSA says the operator detects this by comparing the displayed patrol figure with the speedometer.

What is the shadowing effect in police radar?

Shadowing is a moving-radar error in which the device takes its own-vehicle reference from a slower vehicle traveling in the same direction instead of from the road. NHTSA's SMD Glossary describes the result as a patrol reading lower than actual, which makes the computed target reading too high. Checking the patrol figure against the speedometer catches it.

Does cosine error affect lidar?

Yes, but always in the driver's favor. NHTSA's Lidar Participant Manual states that the cosine effect always works to the motorist's advantage with lidar and does not become a factor until the angle reaches about 10 degrees. Lidar is used only from a stationary position, so the moving-radar exception does not apply.

Cosine error in police radar and lidar