Question

14

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.

asked Mar 29, 2015 at 17:26

6

Answer 1

16

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

enter image description here

answered Mar 29, 2015 at 17:47

Answer 2

13

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

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

answered Mar 29, 2015 at 17:29

4

Answer 3

2

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

answered Mar 29, 2015 at 17:32

Answer 4

2

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.

answered Sep 17, 2020 at 9:30

1

Answer 5

1

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)

answered Jun 19, 2018 at 19:24

Answer 6

0

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

answered Mar 4, 2018 at 20:44

You must log in to answer this question.