COT

Returns the cotangent of the given angle (in radians).

Syntax:

COT(angle)

returns the (trigonometric) cotangent of angle, the angle in radians.To return the cotangent of an angle in degrees, use the RADIANS function.The cotangent of an angle is equivalent to 1 divided by the tangent of that angle.

Example:

COT(PI()/4)

returns 1, the cotangent of PI/4 radians.

COT(RADIANS(45))

returns 1, the cotangent of 45 degrees.

Application:

An application of the cotangent function, often abbreviated as cot, is in the field of civil engineering, specifically when calculating the slope of a ramp or the angle of repose for a pile of granular material.

Let's consider the calculation of the angle of a ramp for a wheelchair. The Americans with Disabilities Act (ADA) has specific guidelines for ramp construction to ensure accessibility. A common standard is a maximum slope of 1:12, which means for every 12 units of horizontal distance, the ramp rises 1 unit vertically.

In this scenario, the cotangent function is a direct way to find the angle of the ramp. The cotangent of an angle in a right-angled triangle is defined as the ratio of the adjacent side to the opposite side.

Opposite side: The vertical rise of the ramp.
Adjacent side: The horizontal run of the ramp.

Given the ADA standard of a 1:12 slope:

Adjacent side = 12
Opposite side = 1

The cotangent of the angle (θ) is: cot(θ)=Adjacent/Opposite=12/1=12

To find the angle θ itself, you would use the inverse cotangent function, often written as arccot or cot−1.

θ=arccot(12)≈4.76∘

This means the ramp has an angle of approximately 4.76 degrees, which is a safe and accessible slope.

Here is a table showing how the cotangent function relates to different ramp slopes:

	Slope (Rise:Run)	Opposite Side (Rise)	Adjacent Side (Run)	cot(θ)=Rise/Run	Angle θ (arccot(cot(θ)))
	A	B	C	D	E
1	1:12	1	12	12/1 = 12	4.76°
2	1:10	1	10	10/1 = 10	5.71°
3	1:8	1	8	8/1 = 8	7.13°
4	1:20	1	20	20/1 = 20	2.86°

This table demonstrates how a steeper ramp (a smaller run-to-rise ratio, like 1:8) results in a smaller cotangent value and a larger angle, while a more gradual ramp (a larger run-to-rise ratio, like 1:20) results in a larger cotangent value and a smaller angle. Civil engineers use this relationship to design ramps that comply with safety and accessibility regulations.