LAC01 Trig Identities MRG 2012.0523

3470 days ago by calcpage123

#0a)Arithmetic show(2+7) show(2-7) show(2*7) show(2/7.n(digits=100)) show(2//7) show(2%7) show(2**7) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}9
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\newcommand{\Bold}[1]{\mathbf{#1}}0.2857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857
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\newcommand{\Bold}[1]{\mathbf{#1}}14
\newcommand{\Bold}[1]{\mathbf{#1}}0.2857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857
\newcommand{\Bold}[1]{\mathbf{#1}}0
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\newcommand{\Bold}[1]{\mathbf{#1}}128
#0b)Algebra show(expand((x+2)^2)) show(factor(x^2+4*x+4)) show(((x^2+4*x+4)/(x+2)).simplify_rational()) solve(x^2+4*x+4==0,x) y=diff(x^2+4*x+4,x) show(y) show(y(x=1)) show(integrate(x^2+4*x+4,x)) show(integrate(x^2+4*x+4,x,0,1).n()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} + 4 \, x + 4
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x + 2\right)}^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}x + 2
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, x + 4
\newcommand{\Bold}[1]{\mathbf{#1}}6
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{3} \, x^{3} + 2 \, x^{2} + 4 \, x
\newcommand{\Bold}[1]{\mathbf{#1}}6.33333333333333
\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} + 4 \, x + 4
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x + 2\right)}^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}x + 2
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, x + 4
\newcommand{\Bold}[1]{\mathbf{#1}}6
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{3} \, x^{3} + 2 \, x^{2} + 4 \, x
\newcommand{\Bold}[1]{\mathbf{#1}}6.33333333333333
#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)) show(f(x)==1) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x\right)^{2} + \cos\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}1
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x\right)^{2} + \cos\left(x\right)^{2} = 1
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x\right)^{2} + \cos\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}1
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x\right)^{2} + \cos\left(x\right)^{2} = 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)) show(bool(h(x)==sec(x)^2)) show(h(x)==sec(x)^2) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(x\right)^{2} + \cos\left(x\right)^{2}}{\cos\left(x\right)^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{\cos\left(x\right)^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(x\right)^{2} + \cos\left(x\right)^{2}}{\cos\left(x\right)^{2}} = \sec\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(x\right)^{2} + \cos\left(x\right)^{2}}{\cos\left(x\right)^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{\cos\left(x\right)^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(x\right)^{2} + \cos\left(x\right)^{2}}{\cos\left(x\right)^{2}} = \sec\left(x\right)^{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)) show(bool(g(x)==csc(x)^2)) show(g(x)==csc(x)^2) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(x\right)^{2} + \cos\left(x\right)^{2}}{\sin\left(x\right)^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{\sin\left(x\right)^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(x\right)^{2} + \cos\left(x\right)^{2}}{\sin\left(x\right)^{2}} = \csc\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(x\right)^{2} + \cos\left(x\right)^{2}}{\sin\left(x\right)^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{\sin\left(x\right)^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(x\right)^{2} + \cos\left(x\right)^{2}}{\sin\left(x\right)^{2}} = \csc\left(x\right)^{2}
#4)Sum of 2 Angles var('y') show(cos(x+y)) show(cos(x+y).simplify_trig()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(x + y\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\sin\left(x\right) \sin\left(y\right) + \cos\left(x\right) \cos\left(y\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(x + y\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\sin\left(x\right) \sin\left(y\right) + \cos\left(x\right) \cos\left(y\right)
#5)Sum of 2 Angles show(cos(x-y)) show(cos(x-y).simplify_trig()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(x - y\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x\right) \sin\left(y\right) + \cos\left(x\right) \cos\left(y\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(x - y\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x\right) \sin\left(y\right) + \cos\left(x\right) \cos\left(y\right)
#6)Sum of 2 Angles show(sin(x+y)) show(sin(x+y).simplify_trig()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x + y\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x\right) \cos\left(y\right) + \sin\left(y\right) \cos\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x + y\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x\right) \cos\left(y\right) + \sin\left(y\right) \cos\left(x\right)
#7)Sum of 2 Angles show(sin(x-y)) show(sin(x-y).simplify_trig()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x - y\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x\right) \cos\left(y\right) - \sin\left(y\right) \cos\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x - y\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x\right) \cos\left(y\right) - \sin\left(y\right) \cos\left(x\right)
#8)Double Angle show(cos(2*x)) show(cos(2*x).simplify_trig()) show(cos(2*x).simplify_trig()+1) show((cos(2*x).simplify_trig()+1)/2) show(plot(cos(x)^2)) show(plot((1+cos(2*x))/2)) show(bool((1+cos(2*x))/2==cos(x)^2)) show((1+cos(2*x))/2==cos(x)^2) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(2 \, x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, \cos\left(x\right)^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, \cos\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, \cos\left(2 \, x\right) + \frac{1}{2} = \cos\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(2 \, x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, \cos\left(x\right)^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, \cos\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, \cos\left(2 \, x\right) + \frac{1}{2} = \cos\left(x\right)^{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).simplify_trig())/2) show(plot(sin(x)^2)) show(plot((1-cos(2*x))/2)) show(bool((1-cos(2*x))/2==sin(x)^2)) show((1-cos(2*x))/2==sin(x)^2) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(2 \, x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, \cos\left(x\right)^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}-2 \, \cos\left(x\right)^{2} + 2
\newcommand{\Bold}[1]{\mathbf{#1}}-\cos\left(x\right)^{2} + 1
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, \cos\left(2 \, x\right) + \frac{1}{2} = \sin\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(2 \, x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, \cos\left(x\right)^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}-2 \, \cos\left(x\right)^{2} + 2
\newcommand{\Bold}[1]{\mathbf{#1}}-\cos\left(x\right)^{2} + 1
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, \cos\left(2 \, x\right) + \frac{1}{2} = \sin\left(x\right)^{2}
#10)Euler's Identity show(taylor(e^x,x,0,5)) show(taylor(e^(i*x),x,0,5)) show(taylor(cos(x),x,0,5)) show(i*taylor(sin(x),x,0,5)) show(bool(e^(i*x)==cos(x)+i*sin(x))) show(e^(i*x)==cos(x)+i*sin(x)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{120} \, x^{5} + \frac{1}{24} \, x^{4} + \frac{1}{6} \, x^{3} + \frac{1}{2} \, x^{2} + x + 1
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{120} i \, x^{5} + \frac{1}{24} \, x^{4} - \frac{1}{6} i \, x^{3} - \frac{1}{2} \, x^{2} + i \, x + 1
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{24} \, x^{4} - \frac{1}{2} \, x^{2} + 1
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{120} i \, x^{5} - \frac{1}{6} i \, x^{3} + i \, x
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(i \, x\right)} = i \, \sin\left(x\right) + \cos\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{120} \, x^{5} + \frac{1}{24} \, x^{4} + \frac{1}{6} \, x^{3} + \frac{1}{2} \, x^{2} + x + 1
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{120} i \, x^{5} + \frac{1}{24} \, x^{4} - \frac{1}{6} i \, x^{3} - \frac{1}{2} \, x^{2} + i \, x + 1
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{24} \, x^{4} - \frac{1}{2} \, x^{2} + 1
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{120} i \, x^{5} - \frac{1}{6} i \, x^{3} + i \, x
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(i \, x\right)} = i \, \sin\left(x\right) + \cos\left(x\right)