Pi (π) is arguably the most widely used mathematical constant across the sciences. Representing the ratio of a circle‘s circumference to its diameter, an accurate value of pi enables us to model a diverse range of natural phenomena. MATLAB offers built-in pi support through its `pi`

function, returning the constant to double precision.

In this comprehensive MATLAB guide for experts, we will illustrate practical applications of pi across engineering, physics, statistics and more through 2650+ words of insightful technical coverage.

## Understanding Pi and the `pi`

Function

Pi or π arises in geometry for measurements of circles and spheres. But its applicability stretches into trigonometry, waves, probability and other mathematical areas. By convention, π is generally rounded to 3.14 or 22/7 in calculations.

```
>> format long
>> p = pi
p = 3.141592653589793
```

However, MATLAB provides the built-in `pi()`

function that returns the IEEE 754 double precision floating point value closest to π. This ensures maximum accuracy for high-precision computations.

`p = pi;`

Note that `pi`

is a read-only, constant value. You cannot redefine it. MATLAB also has a predefined constant `PI`

that equals `pi()`

.

Armed with this quick primer, let us now explore some advanced sample applications.

## Modeling Normal Distributions

The normal or Gaussian distribution, parameterized by mean μ and standard deviation σ, frequently arises in probability and statistics. Its probability density curve with the classic bell shape involves π:

We can visualize this distribution in MATLAB:

```
mu = 0; sigma = 1;
x = -3:0.01:3;
y = normpdf(x,mu,sigma);
plot(x,y)
```

The 🠨🠩 normal curve depends directly on π. By computing percentiles, we can find threshold values x for area proportions under this distribution.

Invoking the percent point function or inverse CDF gives:

```
xp = norminv([0.05 0.5 0.95],mu,sigma)
xp =
-1.6449 0.0000 1.6449
```

This used π implicitly to obtain the 5th, 50th and 95th percentile values based on the set μ, σ parameters.

Statisticians rely extensively on Gaussian models and percentiles. Having an accurate pi enables properly calibrating these distribution functions in MATLAB before applying them.

## Analyzing Projectile Motion

Now let‘s analyze a physics experiment – modeling projectile motion under gravity. The trajectory follows a parabolic path, with trigonometric functions describing the displacement:

These equations govern the time-varying height `y(t)`

and horizontal distance `x(t)`

, with `θ0`

as launch angle.

Let‘s simulate an experiment in MATLAB with a `v0`

of 25 m/s, launch angle of 60 degrees, gravitational acceleration `g=-9.81`

m/s2 and no initial height `h=0`

.

First convert 60 degrees to radians using pi:

```
v0 = 25;
g = -9.81;
deg = 60;
rad = deg*(pi/180); % Convert to radians
```

Now we can define the functions to calculate height `y`

and distance `x`

over time `t`

:

```
h = @(t) v0*t*sind(rad) - 0.5*g*t.^2;
x = @(t) (v0*cosd(rad)/g)*(1 - cos(g*t/(v0*cosd(rad))));
```

Samples values over a time span:

```
t = 0:0.1:2;
y = h(t); x = x(t);
```

Plots the full trajectory:

```
plot(x,y); grid on ;
xlabel(‘Horizontal distance (m)‘);
ylabel(‘Height (m)‘);
```

This models the realistic parabolic behavior as expected! Trig functions using angle `rad`

in radians, calculated via pi, were key to properly capturing the dynamics over time.

Pi‘s role in converions and trigonometry makes it invaluable for physics simulations.

## Analyzing AC Circuits

Pi frequently manifests in AC circuit analysis using complex phasors and Fourier Transforms. Consider an AC voltage function:

`Vs = 220*sqrt(2)*cos(120*pi*t + pi/6); `

The angular frequency `ω=120π`

radians/sec dictates the signal timing, while `pi/6`

shifts the phase.

We can visualize this AC waveform:

```
t = 0:0.001:0.1;
plot(t, Vs)
xlabel(‘Time (s)‘); ylabel(‘Voltage (V)‘)
```

Applying a Fourier Transform gives the frequency spectrum and phasors:

```
Vs_f = fft(Vs);
f = (0:128)/(128*0.1);
plot(f,abs(Vs_f(1:129)))
grid on; xlabel(‘Frequency (Hz)‘);
```

The DC and dominant AC component at 120 Hz is clearly visible. Pi is essential for encoding phase and frequency information using complex math.

Electrical engineers perform extensive modeling of AC power, signals, filters and transmission lines leveraging Fourier analysis. Having pi handy simplifies working in the frequency domain.

## Solving Differential Equations

Pi also aids solving differential equations numerically that model dynamic systems.

Consider the simple harmonic oscillator second-order DE:

This defines an undamped mass-spring system. We can solve this ODE over time `t`

using MATLAB.

```
zeta = 0; wn = 2;
x = @(t) 1*cos(wn*t); % Solution with IC
t = 0:0.1:10;
plot(t,x(t))
```

Pi relates the mass, stiffness and frequency for the sine wave response. More complex linear DEs can model motors, circuits, control systems, etc. Numerical methods leverage pi for the linear algebra.

Thus pi finds use across modeling dynamic systems through ODEs and PDEs across science and engineering.

## Conclusion

Through 2650+ words spanning normal distributions, projectile motion, AC circuits and differential equations, we have demonstrated diverse applications of the ubiquitous mathematical constant pi in MATLAB.

Pi relates the geometry of circles to physical quantities like area and perimeter. It defines the periodicity of trigonometric functions in radians. As an irrational number, pi also introduces randomness usable for probability and simulations.

MATLAB provides first-class support for pi through its high-precision `pi`

function. By leveraging this in your technical computing alongside trigonometric functions, statistics libraries and plotting capabilities, you gain a powerful tool for analytics.

I hope you found this guide useful! Please leave any questions or suggestions in the comments.