Regarding the trigonometric series,
Why does cos start with a 1 and why are the powers of x even numbers
Also for sin, why does sin start with an x and why are the powers odd?
You must be referring to the expansion so sinx and cosx
sinx = x - x^3/3! + x^5/5! - x^7/7! + ... and
cosx = 1 - x^2/2! + x^4/4! - x^6/6! ...
I don't know if you have learned about the Taylor Series expansion.
Here is a page that develops the sinx expansion
http://www.intmath.com/Series-expansion/2_Maclaurin-series.php
The above reference should help you to answer the questions:
Why does cos start with a 1 and why are the powers of x even numbers
- think of what cos(0) equals, and
- is cos(x) an even or odd function?
Also for sin, why does sin start with an x and why are the powers odd?
- Evaluate lim sin(x)/x as x->0
- Is sin(x) an even or odd function?
To understand why the trigonometric series for cosine (cos) and sine (sin) have specific patterns, we need to delve into the properties of these functions and their representations as power series.
Let's start with the cosine function, represented as a power series:
cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ...
The series begins with 1 because when x is 0, the value of cos(x) is 1. Additionally, the powers of x in the terms of the series are even numbers because the cosine function is an even function, meaning that it satisfies the property cos(-x) = cos(x). This property indicates that the function's graph is symmetrical about the y-axis. Since cosine is symmetric, its power series will only contain terms for even powers of x, resulting in an expansion that consists of only even powers of x.
Now, let's consider the sine function represented as a power series:
sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...
The series starts with an x term because, when x is 0, the value of sin(x) is 0. Therefore, the coefficient of the x term is 1. Additionally, the powers of x in the terms of the series are odd numbers because the sine function is an odd function, satisfying the property sin(-x) = -sin(x). This property implies that the graph of sine is also symmetric but with respect to the origin instead of the y-axis. As a result, the power series expansion for sine will only contain terms for odd powers of x.
In summary, the specific patterns observed in the trigonometric series for cosine and sine arise from the properties of these functions and the symmetries they exhibit. The cosine function's series contains terms with even powers of x since it is an even function, while the sine function's series contains terms with odd powers of x since it is an odd function.