Why did God make trigonometry? This makes no sense.

Question

Algebraically solve the equation 2cos²x + 5cosx + 2 = 0 over the domain 0≤x≥2n.... solutions must be in exact values whatever that means.

Answer 1

Question 1:

What do you mean by "0≤x≥2n..."?

Answer:

I assume you mean 0≤x<2π.

Question 2:

How to solve 2cos²x + 5cosx + 2 = 0?

Answer:

First consider and solve the equation

2u² + 5u + 2 = 0

Notice that it is the above equation after replacing cosx with u.

u = (-5 ± √(5² - 4×2×2))/2×2

The two solutions are u = -1/2 and u = -2

Now see which values of x would cause cosx = -1/2 or cosx = -2.

No value of x would give cosx = -2.

In the interval 0≤x<2π there are two values of x that give cosx = -1/2:

x = 2π/3 and x = 4π/3

Question 3:

What does it mean to give exact solution?

Answer:

Keep the solution in the above form as a fraction of π,

rather than approximating it with a decimal representation.

Question 4:

Why did God make trigonometry?

Answer:

I refer to Leopold Kronecker, who seemed to suggest that he knew, when he declared:

"Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk".

That means "The dead God made the whole numbers, everything else was done by people".

Question 5:

Why did people make trigonometry?

Answer:

This is to study waves, which are very useful.

The wave function is very complicated and difficult to deal with.

In contrast, uniform circular motion is very simple to understand --

the length of the arc traversed is simply a linear function of time.

The problem of relating this simple circular motion to the complicated

waves is expressed by trigonometry.

By putting all the complications of waves into one subject of trigonometry,

one can focus on understanding and solving the complications with the abstract

notions of sin and cos.

That is much easier that having to understand those complexities in the

context of waves.