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LAC01 Trig Identities MrG 2012.0530
3298 days ago by lac2012
#0a) Arithmetic show(2+7) show(2-7) show(2*7) show((2/7).n(digits=10)) show((7//2)) show((7%2)) show(2^7)
#0b) Algebra show(expand((x+2)^2)) show(factor(x^2+4*x+4)) show(((x^2+4*x+4)/(x+2)).simplify_rational()) show(solve(x^2+4*x+4==0,x)) f(x)=(diff(x^2+4*x+4,x)) show(f(x)) show(f(-1)) show(integrate(x^2+4*x+4,x)) show(integrate(x^2+4*x+4,x,0,1))
#1) Pythagorean f(x)=sin(x)^2+cos(x)^2 show(f(x)) show(f(x).simplify_trig()) show(plot(f(x),xmin=-2,xmax=2,ymin=-2,ymax=2)) show(bool(f(x)==1))
#2) Pythagorean h(x)=f(x)/cos(x)^2 show(h(x)) show(h(x).simplify_trig()) show(plot(h(x))) show(plot(sec(x)^2))
#3) Pythagorean g(x)=f(x)/sin(x)^2 show(g(x)) show(g(x).simplify_trig()) show(plot(g(x),0.1,3)) show(plot(csc(x)^2,0.1,3))
#4) Sum of 2 Angles var('y') show(cos(x+y)) show(cos(x+y).simplify_trig()) show(plot3d(cos(x+y), (x,-2,2), (y,-2,2)),viewer='tachyon') show(plot3d(cos(x)*cos(y)-sin(x)*sin(y), (x,-2,2), (y,-2,2)),viewer='tachyon')
#5) Sum of 2 Angles show(cos(x-y)) show(cos(x-y).simplify_trig()) show(plot3d(cos(x-y), (x,-2,2), (y,-2,2)),viewer='tachyon') show(plot3d(cos(x)*cos(y)+sin(x)*sin(y), (x,-2,2), (y,-2,2)),viewer='jmol')
7
500
/home/pub/320/cells/7/.jmol_images/sage1-size500.jmol.png?1652844365.68
/home/pub/320/cells/7/sage1-size500.jmol?1652844365.68
/home/pub/320/jsmol
notActivated
#6) Sum of 2 Angles show(sin(x+y)) show(sin(x+y).simplify_trig()) show(plot3d(sin(x+y), (x,-2,2), (y,-2,2)),viewer='tachyon') show(plot3d(sin(x)*cos(y)+sin(x)*cos(y), (x,-2,2), (y,-2,2)),viewer='jmol')
8
500
/home/pub/320/cells/8/.jmol_images/sage1-size500.jmol.png?1652844365.68
/home/pub/320/cells/8/sage1-size500.jmol?1652844365.68
/home/pub/320/jsmol
notActivated
#7) Sum of 2 Angles show(sin(x-y)) show(sin(x-y).simplify_trig()) show(plot3d(sin(x-y), (x,-2,2), (y,-2,2)),viewer='tachyon') show(plot3d(sin(x)*cos(y)-sin(y)*cos(x), (x,-2,2), (y,-2,2)),viewer='jmol')
9
500
/home/pub/320/cells/9/.jmol_images/sage1-size500.jmol.png?1652844365.68
/home/pub/320/cells/9/sage1-size500.jmol?1652844365.68
/home/pub/320/jsmol
notActivated
#8) Double Angle show(cos(2*x)) show(cos(2*x).simplify_trig()) show((1+cos(2*x)).simplify_trig()) show(((1+cos(2*x))/2).simplify_trig()) show(plot((1+cos(2*x))/2)) plot(cos(x)^2)
#9) Double Angle show(cos(2*x)) show(cos(2*x).simplify_trig()) show((1-cos(2*x)).simplify_trig()) show(((1-cos(2*x))/2).simplify_trig()) show(plot((1-cos(2*x))/2)) plot(sin(x)^2)
#10) Euler's Identity show(taylor(e^x,x,0,9)) show(taylor(e^(i*x),x,0,9)) show(taylor(cos(x),x,0,9)) show(i*taylor(sin(x),x,0,9)) show(e^(i*x)==cos(x)+i*sin(x)) show(bool(e^(i*x)==cos(x)+i*sin(x)))
#10a) R^R show((2+0*i)^(3+0*i))
#10b) I^R show((0+2*i)^(3+0*i))
#10c) C^R show((1+i)^(3+0*i)) show(expand((1+i)^0)) show(expand((1+i)^1)) show(expand((1+i)^2)) show(expand((1+i)^3)) show(expand((1+x)^0)) show(expand((1+x)^1)) show(expand((1+x)^2)) show(expand((1+x)^3))
#10d) R^I show((2+0*i)^(0+3*i)) show(((2+0*i)^(0+3*i)).n()) show((e^log(2))^(3*i)) show((e^(3*log(2)*i)).simplify_radical()) show((cos(3*log(2))+i*sin(3*log(2))).n())
#10e) I^I show((0+2*sqrt(3)*i)^(0+3*i)) show(((0+2*sqrt(3)*i)^(0+3*i)).n())
#10f) C^I show((2+2*sqrt(3)*i)^(0+3*i)) show(((2+2*sqrt(3)*i)^(0+3*i)).n())
#10g) R^C show((2+0*i)^(3+3*i)) show(((2+0*i)^(3+3*i)).n())
#10h) I^C show((0+2*sqrt(3)*i)^(3+3*i)) show(((0+2*sqrt(3)*i)^(3+3*i)).n())
#10i) C^C show((2+2*sqrt(3)*i)^(3+3*i)) show(((2+2*sqrt(3)*i)^(3+3*i)).n())