ATAN

Returns the inverse tangent (the arctangent) of a number.

Syntax:

ATAN(number)

returns the inverse trigonometric tangent of number, in other words the angle (in radians) whose tangent is number. The angle returned is between -PI/2 and PI/2.To return the angle in degrees, use the DEGREES function.

Example:

ATAN(1)

returns 0.785398163397448 (PI/4 radians).

DEGREES(ATAN(1))

returns 45. The tangent of 45 degrees is 1.

Application:

Calculation the Launch Angle of a Projectile

Imagine a catapult launching a projectile. We know the horizontal distance the projectile travels (x) and the maximum vertical height it reaches (y). We can use the ATAN function to calculate the launch angle (θ) of the projectile.

In physics, the tangent of the launch angle is the ratio of the vertical component of the initial velocity (vy) to the horizontal component (vx). While we don't have the velocities directly, we can use the ratio of the vertical distance to the horizontal distance to approximate the initial angle. A simplified formula for the launch angle is:

θ=arctan(y/x)

where:

θ is the launch angle in radians (we can convert this to degrees later).
y is the maximum vertical height.
x is the horizontal distance to the point where the maximum height is reached.

Example Data

Let's say we have three different test launches with the following data:

	Launch	Horizontal Distance (x)	Maximum Height (y)
	A	B	C
1	1	20 meters	15 meters
2	2	30 meters	10 meters
3	3	25 meters	25 meters

Calculation using ATAN

Now, let's use the formula to calculate the launch angle for each launch.

Launch 1:

Ratio: y/x=15/20=0.75
Angle (in radians): θ=arctan(0.75)≈0.6435 radians
Angle (in degrees): θ=0.6435×180/π≈36.87°

Launch 2:

Ratio: y/x=10/30=0.333...
Angle (in radians): θ=arctan(0.333...)≈0.3218 radians
Angle (in degrees): θ=0.3218×180/π≈18.43°

Launch 3:

Ratio: y/x=25/25=1
Angle (in radians): θ=arctan(1)=π/4≈0.7854 radians
Angle (in degrees): θ=0.7854×180/π=45°

Results Table

Here is the completed table, including the calculated launch angles:

	Launch	Horizontal Distance (x)	Maximum Height (y)	Ratio (y/x)	Launch Angle (radians)	Launch Angle (degrees)
	A	B	C	D	E	F
1	1	20 meters	15 meters	0.75	0.6435	36.87°
2	2	30 meters	10 meters	0.333333333	0.3218	18.43°
3	3	25 meters	25 meters	1	0.7854	45.00°

This example demonstrates how the ATAN function is a crucial tool for converting the ratio of two physical measurements into a meaningful angle, which is essential for understanding and predicting the trajectory of a projectile.