Understanding Curves: f(x) = sin(x), g(x) = cos(x)

[Dr-NiKoN]

Not sure if it's called "curves" in English, but what I am referring to is graphs that repeat over a given time.
ie
f(x) = sin(x)

The problem I am having is understanding the following:
Given:
f(x) = sin(x)
g(x) = cos(x)

Find f(x) - g(x) by A(cos(x - x0)).
Which gives:
A(cos(x - x0)) = A(cos(x0))cos(x) + A(sin(x0))sin(x)
A(cos(x0)) = 1
A(sin(x0)) = -1

thus:
A = sqrt(1^2 + (-1)^2) = 1

Finding x0:
tan wx0 = 1/(-1)
1*x0 = arctan(-1)
x0 = -0.79

Here I'm pretty much lost. I've probably done some mistakes along the way as well :(
What is -0.79? x0 is supposed to be the 'top' of the curve right?
ie to find every top, you would have something like:
x0 +/- |n|*2PI

Where n is a whole number(1..inf) and 2PI would be the period of each "curve".

Could someone explain this to me, I'm trying to learn this by just reading a book and I'm having a hard time.

 

[Dr-NiKoN]

thus:
A = sqrt(1^2 + (-1)^2) = 1

Brainfart.
sqrt(2) is sqrt(2), not 1

 

[M98Ranger]

First of all, yes, the term "curves" is commonly used in English to refer to graphs that repeat over a given time. In mathematics, these are also known as periodic functions.

Now, let's break down the problem step by step. We are given two functions, f(x) = sin(x) and g(x) = cos(x), and we are asked to find the difference between them, which is denoted by f(x) - g(x). To do this, we need to use the given formula A(cos(x - x0)) and solve for A and x0.

First, let's focus on finding A. We are given that A(cos(x0)) = 1 and A(sin(x0)) = -1. These equations come from the fact that cos(x - x0) = cos(x0)cos(x) + sin(x0)sin(x), which is a trigonometric identity. By substituting the given values of A(cos(x0)) and A(sin(x0)), we can solve for A.

A(cos(x0)) = 1
1 = cos(x0)
x0 = cos^-1(1)
x0 = 0

Similarly, we can solve for x0 in the second equation.

A(sin(x0)) = -1
-1 = sin(x0)
x0 = sin^-1(-1)
x0 = -π/2

However, we need to find the value of x0 that satisfies both equations, which is why we set them equal to each other.

x0 = cos^-1(1) = sin^-1(-1)
x0 = 0 = -π/2

Since these two values are not equal, we can conclude that there is no value of x0 that satisfies both equations simultaneously. Therefore, there is no single value of x0 that represents the "top" of the curve.

Instead, we can think of x0 as a range of values that represent the "tops" of the curve. As you mentioned, x0 can be written as x0 +/- n*2π, where n is a whole number. This is because the curve repeats itself every 2π units.

For example, if we take x0 = 0, we get the "top" of the curve at x = 0. But if we take x0 = 2π, we get the "top" of the curve at x