What is cos exactly?

Mathematicians discovered certain patterns while performing geometric calculations. For example, the number pi (π) is always found by dividing the circumference of a circle by its diameter, and regardless of the size of the circle, this ratio remains constant. Since this ratio is used repeatedly in calculations, it was given a special name and represented in formulas as such: π=3.141...\pi = 3.141...π=3.141....

Just as we arrived at this universal rule, sine (sin⁡) and cosine (cos⁡) also went through a similar process. It was observed that in triangles of different sizes with the same angles, the ratio of the opposite side to the hypotenuse (sin⁡) and the adjacent side to the hypotenuse (cos⁡) always remained the same. These consistently fixed ratios were also given specific names.

In this way, since we consistently arrive at the same ratios, the concept of cosine (cos) was developed to be used alongside specific angles. Similarly, the ratio of the opposite side to the hypotenuse will always be the same, regardless of the size of the triangles. After establishing this standard, we can state that, for all cases, cos(60°) = 0.5. With this approach, we actually know the sine (sin) and cosine (cos) values for all angles. These values have been compiled into what is known as a trigonometric table.

These values will always remain the same for the given angles, regardless of the lengths of the triangles. These ratios will not change.

So, where else do we use the cosine (cos) value?

Let's look at the unit circle, which is a circle with a radius of 1 (Image 4). Here, thanks to cosine (cos), we can determine the x and y components of the given pink line. In other words, we can find the components of the vector.

So, what happens if we multiply a vector by cos(θ)?

For example, let's consider these two vectors in the x,y plane (Image 5). Let's connect their starting points.

We have connected their starting points, and you can notice the angle formed between them (Image 6). In the next image, I will refer to this angle as θ.

It looks like we are heading towards something familiar.

Now, let's draw a perpendicular line from the endpoint of vector a to line b (Image 7).

Here, we have formed a right triangle, and we know that the value k can be found by multiplying the magnitude of a by cos(θ). We call this the projection of vector a onto vector b. The reason we do this is to determine how much vector a is aligned with vector b.

These informations show that cosine is not a concept that appeared out of nowhere; rather, it is simply a basic ratio. I wanted to explain how certain calculations are performed using this ratio. In the future, understanding concepts like the dot product or cross product, which are crucial in fields like game and graphics programming, will become much easier.