#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)
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#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))
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#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))
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#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))
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#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))
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#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')
 
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#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')
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#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')
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#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')
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#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)
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#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)
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#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)))
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#10a) R^R
show((2+0*i)^(3+0*i))
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#10b) I^R
show((0+2*i)^(3+0*i))
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#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))
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#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())
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#10e) I^I
show((0+2*sqrt(3)*i)^(0+3*i))
show(((0+2*sqrt(3)*i)^(0+3*i)).n())
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#10f) C^I
show((2+2*sqrt(3)*i)^(0+3*i))
show(((2+2*sqrt(3)*i)^(0+3*i)).n())
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#10g) R^C
show((2+0*i)^(3+3*i))
show(((2+0*i)^(3+3*i)).n())
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#10h) I^C
show((0+2*sqrt(3)*i)^(3+3*i))
show(((0+2*sqrt(3)*i)^(3+3*i)).n())
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#10i) C^C
show((2+2*sqrt(3)*i)^(3+3*i))
show(((2+2*sqrt(3)*i)^(3+3*i)).n())
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