Imagine you stand on a distance, say \( 10 \) m away, watching someone throwing a ball upwards. A straight line from you to the ball will then make an angle with the horizontal that increases and decreases as the ball goes up and down. Let us consider the ball at a particular moment in time, at which it has a height of \( 10 \) m.
What is the angle of the line then? Again, this could easily be done with a calculator, but we continue to address gentle mathematical problems when learning to program. Before thinking of writing a program, one should always formulate the algorithm, i.e., the recipe for what kind of calculations that must be performed. Here, if the ball is \( x \) m away and \( y \) m up in the air, it makes an angle \( \theta \) with the ground, where \( \tan\theta = y/x \). The angle is then \( \tan^{-1}(y/x) \).
Let us make a Matlab program for doing these calculations. We introduce
names x and y for the position data \( x \) and \( y \), and the
descriptive name angle for the angle \( \theta \).
The program is stored in a file
ball_angle.m:
x = 10; % Horizontal position
y = 10; % Vertical position
angle = atan(y/x);
(angle/pi)*180 % Computes and prints to screen
Before we turn our attention to the running of this program, let us
take a look at one new thing in the code. The line angle = atan(y/x),
illustrates how the function atan, corresponding to \( \tan^{-1} \)
in mathematics, is called with the
ratio y/x as input parameter or argument.
The atan function takes one argument, and the computed value is
returned from atan. This means that where we see atan(y/x),
a computation is performed (\( \tan^{-1}(y/x) \)) and the result "replaces"
the text atan(y/x). This is actually no more magic than if we
had written just y/x: then the computation of y/x would take place,
and the result of that division would replace the text y/x.
Thereafter, the result is assigned to the name angle
on the left-hand side of =.
Note that the trigonometric functions, such as atan, work with
angles in radians. The return value of atan must hence be converted
to degrees, and that is why we perform the computation (angle/pi)*180.
With the missing semi-colon, Matlab will do the computations and
print the result to the screen. And yes, the famous
pi (\( \pi \)) is a variable that is known
to Matlab.