# Revision history [back]

### simplify trig, abs, and sqrt expression

I'm computing the Frenet frame for a helix. Sage does not seem to want to simplify expressions like $\sqrt{|r^2 \sin(\theta)|^2+|r^2 \cos(\theta)^2|}$ without using simplify_full followed by simplify_trig. My code is below. What I've done is not elegant. Can anyone suggest a cleaner approach?

var('r,theta,x,y,t,R')
assume(r>0)
assume(R>0)
assume(t,'real')
f(t) = (R*cos(t),R*sin(t),t)
tangent=diff(f(t),t)
normal=diff(tang,t)
binormal=tangent.cross_product(normal)
norm_of_tangent=tangent.norm().simplify_full().simplify_trig()
norm_of_normal=normal.norm().simplify_full().simplify_trig()
norm_of_binormal=binormal.norm().simplify_full().simplify_trig()
F=matrix([tangent/norm_of_tangent,normal/norm_of_normal,binormal/norm_of_binormal]).transpose()
F


The result is:

[-R*sin(t)/sqrt(R^2 + 1), -cos(t), sin(t)/sqrt(R^2 + 1)]
[R*cos(t)/sqrt(R^2 + 1), -sin(t), -cos(t)/sqrt(R^2 + 1)]
[1/sqrt(R^2 + 1), 0, (R^2*sin(t)^2 + R^2*cos(t)^2)/(sqrt(R^2 + 1)*R)]


### simplify trig, abs, and sqrt expression

I'm computing the Frenet frame for a helix. Sage does not seem to want to simplify expressions like $\sqrt{|r^2 \sin(\theta)|^2+|r^2 \cos(\theta)^2|}$ without using simplify_full followed by simplify_trig. My code is below. What I've done is not elegant. Can anyone suggest a cleaner approach?

var('r,theta,x,y,t,R')
assume(r>0)
assume(R>0)
assume(t,'real')
f(t) = (R*cos(t),R*sin(t),t)
tangent=diff(f(t),t)
normal=diff(tang,t)
normal=diff(tangent,t)
binormal=tangent.cross_product(normal)
norm_of_tangent=tangent.norm().simplify_full().simplify_trig()
norm_of_normal=normal.norm().simplify_full().simplify_trig()
norm_of_binormal=binormal.norm().simplify_full().simplify_trig()
F=matrix([tangent/norm_of_tangent,normal/norm_of_normal,binormal/norm_of_binormal]).transpose()
F


The result is:

[-R*sin(t)/sqrt(R^2 + 1), -cos(t), sin(t)/sqrt(R^2 + 1)]
[R*cos(t)/sqrt(R^2 + 1), -sin(t), -cos(t)/sqrt(R^2 + 1)]
[1/sqrt(R^2 + 1), 0, (R^2*sin(t)^2 + R^2*cos(t)^2)/(sqrt(R^2 + 1)*R)]