$\begingroup$

I needed the PI constant in C++, and I was lead to the answer that:

const PI = atan(1) * 4

Note that despite involving code, I'm asking this from a mathematics perspective.

I have 2 questions about this:

• Is this an estimation of PI, or should it give me a large degree of accuracy?
• How does this give PI?

On the second point, if I understand atan correctly, it takes a ratio of two sides, and returns the corresponding angle.

That means that atan(1) is referring a situation with 2 equal length sides.

I think I'm misunderstanding something though, as atan(1) gives me 0.7853981633974483, which seems like a very tiny angle.

If anyone can fill in the holes, it would be greatly appreciated.

$\endgroup$ 6

6 Answers

$\begingroup$

I'm including this little gif from Wikipedia as a great way to understand radians.

$\endgroup$ $\begingroup$

The function $\arctan\colon \mathbb{R}\to (-\frac{\pi}{2},\frac{\pi}{2})$ is the inverse of $\tan$. (for the right domain of definition). As $\tan \frac{\pi}{4} = 1$, this means that $\arctan 1=\frac{\pi}{4}$.

Regarding your question about angles: angles are (in mathematics) measured in radians (in $[0,2\pi)$ or $[-\pi,\pi)$), not in degrees: you should expect a value or order $\pi$ or so, not ranging between $0$ and $360$.

$\endgroup$ 4 $\begingroup$

You certainly know that $\sin{\frac{\pi}{4}}=\cos{\frac{\pi}{4}}=\frac{\sqrt{2}}{2}$ so one has $\tan{\frac{\pi}{4}}=1$ and therefore $\pi=4\tan^{-1}{1}$

$\endgroup$ $\begingroup$

Math explanation from a non-math person:

In a right angled triangle if the two short sides are equal, the angle is 45 degrees.

45 degrees in radians is π/4. (The full circumference is 2πr, 180 degrees is π and 45 degrees is π/4)

sin π/4 = cos π/4 because the two sides are equal.

tan π/4 = tan 45 = 1.

Arctan(1) is the degree (or radian) which returns a value of 1. So arctan of 1 is either 45 degrees or π/4.

π = 4*arctan(1)

$\endgroup$ $\begingroup$

This shows geometric explanation for relationship between tan, atan and Pi.

Because horizontal segment AB = 1 and vertical segment BD = 1, angle alpha = 45°. From there you can use atan( BD ) to determine 45° in radiant and take that times 4 to get Pi.

$\endgroup$ $\begingroup$

You are correct, in all numeric programing languages like fortran, c, c++, and many others, for a program generalized input line that looks something like:

 print, numeric, %pi , acos(-1), 4*atan(1)

all return the same numeric value. Here the system stored value %pi may be faster than trig evaluation.

 3.141592653589793, 3.141592653589793, 3.141592653589793

$\endgroup$

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy