HOW TO FIX PI

Let's say that for some reason, you are in a chat with a robot which has redefined all of math. For instance,

• pi = "fucking"
  
• e = 3
  
• sin = infinity
  
• cos = [3, 4.33, "meowing"]
  
• exp = ["e", "x", "p"]
  
• infinity = 2
  
• NaN = 10
  
• false = true
  
• OR JUST ANYTHING = "all star"

Thankfully, all the numbers work and so do all the basic functions. How then, can you fix (some of) mathematics? Fear not, with this simple guide, you too, can fix a large portion of important math.

1. pi=4*(.5!)^2
   
2. sqrt(x)=x^.5
   
3. i=sqrt(-1)
   • (OPTIONAL, if re isn't screwed up.) i=i-re(i)
4. e=(i^i)^(-2/pi)
   
5. exp(x)=e^x
   
6. sin(x)=(exp(i*x)-exp(-i*x))/(2*i), cos(x)=.5*(exp(i*x)+exp(-i*x))
   
7. tan(x)=sin(x)/cos(x), cot(x)=cos(x)/sin(x), sec(x)=1/cos(x), csc(x)=1/sin(x)
   • Assuming you have atan2 and it isn't screwed up (or you have arg, which is the same), you can fix all inverse trig
8. atan(x)=atan2(x,1) OR atan(x)=arg(1+x*i)
   
9. asin(x)=atan(x/sqrt(1-x^2)), acos(x)=atan(sqrt(1-x^2)/x)
   
10. acot(x)=atan(1/x), asec(x)=acos(1/x), acsc(x)=asin(1/x)
    • If you have any inverse hyperbolic function or the natural logarithm, you can get all these functions back too:
11. asinh(x)=acosh(sqrt(x^2+1)), acosh(x)=asinh(sqrt(x^2-1))
    
12. atanh(x)=asinh(x/sqrt(1-x^2)) OR acosh(1/sqrt(1-x^2))
    
13. asinh(x)=atanh(x/sqrt(1+x^2)), acosh(x)=atanh(sqrt(x^2-1)/x)
    
14. log(x)=asinh((x^2-1)/(2*x)) OR acosh((x^2+1)/(2*x)) OR atanh((x^2-1)/(x^2+1))
    
15. atanh(x)=.5*log((1+x)/(1-x)), acosh(x)=.5*log((x+1)/(x-1))