How does pi relate to sine and consine?

[girlsnguitars]

I know pi/2 = 90 degrees and that it's related but how do I express this?

 

[chroot]

A circle with a radius of one unit has a circumference of 2 pi units ("radians"). If you consider the same circumference in degrees, you reach the equality:

2 pi radians = 360 degrees
pi / 2 radians = 90 degrees

- Warren

 

[cesiumfrog]

The number of degrees in a circle is historical, but choosing radians uniquely simplifies the expression of gradient of a sinusoid.

 

[tiny-tim]

Welcome to PF!

I know pi/2 = 90 degrees and that it's related but how do I express this?

Hi girlsnguitars! Welcome to PF!

(have a pi: π and a degrees: º )

π/2, π/4, and so on, are related to arc-length rather than to sin or cosine …

the length of a 90º arc is πr/2, and of a 45º arc is πr/4, and so on …

 

[HallsofIvy]

tiny-tim makes a very good point. When, in pure mathematics (as opposed to applications of mathematics), we talk about the functions sin(x) and cos(x), the x is not, and cannot be, angles- and so not measured in degrees or radians.

I have sometimes expressed this as follows:

Suppose, on a test, a problem defines f(x)= x² and asks for f(3). Naturally, you take out your calculator, enter the number 3, press the "square" key, and get 9! Now, the very next problem defines g(x)= sin(x) and asks for g(3). Since you already have your calculator out, you press the "sine" key, enter the number 3 again, and get 0.0623359...
However, when you get the test back you find the first problem has been marked correct, the second wrong! Your teacher tells you that you should have had your calculator in "radian" mode, not "degree". But the problem didn't say "3 radians" (or "3 degrees"). Should you go to the chair of the department and complain? The president of the college?

Yes, the problem did not say "3 radians", just as the first problem didn't say "3 degrees" or "3 feet" or "3 Joules", etc. These are mathematical functions- there are NO units. That is one difficulty with the usual "right triangle" definition of sine and cosine (in addition to the obvious fact that you can only apply them to angles between 0 and 90 degrees). There are other, better, definitions of sine and cosine. The most commonly used is the "circular definition" where sin(t) and cos(t) are defined as the x and y coordinates of the point on the unit circle a distance t, measured along the circumference, from (1, 0). That is the definition tiny-tim is referring to. It is, in my opinion, flawed itself (that's why I used the term "circular definition"- that's a joke, son!)

Another way to define sine and cosine are as solutions to the differential equation y"= -y, with initial conditions y(0)= 0, y'(0)= 1 (for sine) and y(0)= 1, y'(0)= 0 (for cosine).

Yet another is to define sin(x) as the infinite sum
\sum_{n= 0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1}
and define cos(x) as the infinite sum
\sum_{n= 0}^\infty \frac{(-1)^n}{(2n)!} x^{2n}