Reminiscing about past pre-calculus knowledge, let’s take a look at the sine and cosine values of angles that we have come to know and love:
Some of you may have noticed a pattern in these values, which looks like 
for sine and 
for cosine. Unfortunately, this pattern stops at 1, because sine and cosine values cannot exceed 1. Wouldn’t it be nice, though, if the values continued to be 
etc? This is not possible using the same unit circle, which is why logarithmic spirals come into play. More specifically, the Fibonacci Spiral.
In 1202, Leonardo of Pisa, best known as Fibonacci, reintroduced a sequence to Western Culture in which each term was the sum of the two previous terms. Starting at 1, the simplest Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … and so on. If we build squares with sides that are Fibonacci numbers and place them next to each other as in the figure below, the spiral resulting from drawing quarter circles in each square is the Fibonacci spiral, also known as the golden spiral.
In this spiral, each quarter term can be seen as a quarter unit circle, and can be assigned different angles and values. Since the circles have a connection- every radius being the sum of the two previous ones- the trigonometric values we assign to one of them, should be relatable to the rest. Verifying this relationship is the main objective of this research project. After learning all there is to know about the properties and applications of the Fibonacci Spiral, the goal is to create a new spiral-based trigonometry using the golden spiral.
