ACOS

Returns the inverse cosine (the arccosine) of a number.

Syntax:

ACOS(number)

returns the inverse trigonometric cosine of number, in other words the angle (in radians) whose cosine is number. The angle returned is between 0 and PI.To return the angle in degrees, use the DEGREES function.

Example:

ACOS(-1)

returns 3.14159265358979 (PI radians).

DEGREES(ACOS(0.5))

returns 60. The cosine of 60 degrees is 0.5.

Application:

Calculating the Angle of a Satellite's Orbit

Imagine you are a satellite engineer. You have a satellite orbiting the Earth. You know its position in a 2D coordinate system relative to the Earth's center, and you want to determine the angle of its position relative to a reference point (e.g., the positive x-axis).

Let's say the satellite's position is given by the coordinates (x,y). The distance of the satellite from the center of the Earth (the radius of its orbit) is r= $x^{2} + y^{2}$ .

From basic trigonometry, we know that the cosine of the angle θ is given by:

cos(θ)=x/r

To find the angle θ, you would use the ACOS function:

θ=ACOS(x/r)

The result will be in radians, which you can then convert to degrees if needed (multiply by 180/π).

Example Scenario:

A satellite is in orbit. Its position at three different points in time is recorded.

Point 1: Coordinates are (12000,5000) km.
Point 2: Coordinates are (−8000,10000) km.
Point 3: Coordinates are (0,13000) km.

We can use the ACOS function to find the angle of the satellite at each point.

Table: Satellite Position and Calculated Angle

	Point	x (km)	y (km)	r (km) = $x^{2} + y^{2}$	x/r	θ (radians) = ACOS(x/r)	θ (degrees)
	A	B	C	D	E	F	G
1	1	12000	5000	13000	0.923	0.394	22.6°
2	2	-8000	10000	12806	-0.625	2.245	128.6°
3	3	0	13000	13000	0	1.571	90°

This example shows how the ACOS function is a fundamental tool for solving problems in physics, engineering, and other fields that involve angles and coordinate systems. It allows us to reverse the cosine operation to find the underlying angle.