MAO 1 prise en main.sagews

1+1

2

%typeset_mode True

pi+pi

$\displaystyle 2 \, \pi$

Je suis une super Cpi 1

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "<string>", line 1 Je suis une super Cpi Integer(1) ^ SyntaxError: invalid syntax

# je suis une super CPI 1

%md ***je suis une super CPI 1***

je suis une super CPI 1

%md $\sum_{k=0}^{n}k^2$

$\sum_{k=0}^{n}k^2$

var('a b')
expand((a+b)^3)

($\displaystyle a$, $\displaystyle b$)

$\displaystyle a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}$

((a+b)^3).expand().factor()

$\displaystyle {\left(a + b\right)}^{3}$

var('x')

$\displaystyle x$

f(x)==tan(x)

$\displaystyle \tan\left(x\right) = \tan\left(x\right)$

solve{(cos(x)==y),x,((y==0),y)}

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "<string>", line 1 solve{(cos(x)==y),x,((y==Integer(0)),y)} ^ SyntaxError: invalid syntax

var('x')

x

solve(sin(x)/cos(x),x)

[x == 0]

f(x)=tan(x)

solve(cos(x)==0,x)

[x == 1/2*pi]

.simplify_trig(tan(x),x)

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 996, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "<string>", line 1 .simplify_trig(tan(x),x) ^ SyntaxError: invalid syntax

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 996, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "<string>", line 1 var'x' ^ SyntaxError: invalid syntax

var('x')
f(x)=tan(x)
f.simplify_trig(f(-x))

x x |--> sin(x)/cos(x)

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 996, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "<string>", line 1 var'x' ^ SyntaxError: invalid syntax

var('x')
f(x)=tan(x)
f.simplify_trig(f(pi+x))

x x |--> sin(x)/cos(x)

diff(f)

x |--> tan(x)^2 + 1

(tan(x)^2+1).simplify_trig(tan(x)^2+1)

cos(x)^(-2)

f(pi/6)

$\displaystyle \frac{1}{3} \, \sqrt{3}$

n(1/3*sqrt(3))

$\displaystyle 0.577350269189626$

f(pi/4)

$\displaystyle 1$

%typeset_mode True

f(pi/3)

$\displaystyle \sqrt{3}$

n(sqrt(3))

$\displaystyle 1.73205080756888$

limit(f(x),x=-pi/2,dir='plus')

$\displaystyle -\infty$

limit(f(x),x=pi/2,dir='minus')

$\displaystyle +\infty$

f(x).integral(f(x))

$\displaystyle -\frac{1}{2 \, {\left(\sin\left(x\right)^{2} - 1\right)}}$

var('x')
f(x)=tan(x)
plot(f(x))

x

plot(f(x),-pi/2,pi/2,ymin=-10,ymax=10)

var('x')
f(x)=tan(x)
plot(f(x),-2*pi,2*pi,ymin=-10,ymax=10)

x

plot(f(x),-2*pi,2*pi,ymin=-10,ymax=10,detect_poles='show')

%typeset_mode True

var('x')
f(x)=(e^x+e^(-x))/2
f.simplify_trig(f(pi+x))
f.simplify_trig(f(-x))
diff(f(x))
plot(f(x))
f(x).integral(f(x))

$\displaystyle x$

$\displaystyle x \ {\mapsto}\ \frac{1}{2} \, {\left(e^{\left(2 \, x\right)} + 1\right)} e^{\left(-x\right)}$

$\displaystyle x \ {\mapsto}\ \frac{1}{2} \, {\left(e^{\left(2 \, x\right)} + 1\right)} e^{\left(-x\right)}$

$\displaystyle -\frac{1}{2} \, e^{\left(-x\right)} + \frac{1}{2} \, e^{x}$

$\displaystyle \frac{1}{8} \, {\left(e^{\left(-x\right)} - e^{x}\right)}^{2}$

%typeset_mode True

var('x')
f(x)=(e^x-e^(-x))/2
f.simplify_trig(f(pi+x))
f.simplify_trig(f(-x))
diff(f(x))
plot(f(x))
f(x).integral(f(x))

$\displaystyle x$

$\displaystyle x \ {\mapsto}\ \frac{1}{2} \, {\left(e^{\left(2 \, x\right)} - 1\right)} e^{\left(-x\right)}$

$\displaystyle x \ {\mapsto}\ \frac{1}{2} \, {\left(e^{\left(2 \, x\right)} - 1\right)} e^{\left(-x\right)}$

$\displaystyle \frac{1}{2} \, e^{\left(-x\right)} + \frac{1}{2} \, e^{x}$

$\displaystyle \frac{1}{8} \, {\left(e^{\left(-x\right)} - e^{x}\right)}^{2}$

var('a')
var('b')
cos(a+b).expand_trig(cos(a+b))
sin(a+b).expand_trig()
tan(a+b).expand_trig()

$\displaystyle a$

$\displaystyle b$

$\displaystyle \cos\left(a\right) \cos\left(b\right) - \sin\left(a\right) \sin\left(b\right)$

$\displaystyle \cos\left(b\right) \sin\left(a\right) + \cos\left(a\right) \sin\left(b\right)$

$\displaystyle -\frac{\tan\left(a\right) + \tan\left(b\right)}{\tan\left(a\right) \tan\left(b\right) - 1}$

(cos(a)^2+sin(a)^2).simplify_trig()

$\displaystyle 1$

cos(2*a).expand_trig()
cos(2*a).simplify_trig()

$\displaystyle \cos\left(a\right)^{2} - \sin\left(a\right)^{2}$

$\displaystyle 2 \, \cos\left(a\right)^{2} - 1$

sin(2*a).expand_trig()
sin(2*a).simplify_trig()

$\displaystyle 2 \, \cos\left(a\right) \sin\left(a\right)$

$\displaystyle 2 \, \cos\left(a\right) \sin\left(a\right)$

tan(2*a).expand_trig()
tan(2*a).simplify_trig()

$\displaystyle -\frac{2 \, \tan\left(a\right)}{\tan\left(a\right)^{2} - 1}$

$\displaystyle \frac{2 \, \cos\left(a\right) \sin\left(a\right)}{2 \, \cos\left(a\right)^{2} - 1}$

cos(a+pi).simplify_trig()
cos(a-pi).simplify_trig()
cos(a+pi/2).simplify_trig()
cos(a-pi/2).simplify_trig()

$\displaystyle -\cos\left(a\right)$

$\displaystyle -\cos\left(a\right)$

$\displaystyle -\sin\left(a\right)$

$\displaystyle \sin\left(a\right)$

sin(a+pi).simplify_trig()
sin(a-pi).simplify_trig()
sin(a+pi/2).simplify_trig()
sin(a-pi/2).simplify_trig()

$\displaystyle -\sin\left(a\right)$

$\displaystyle -\sin\left(a\right)$

$\displaystyle \cos\left(a\right)$

$\displaystyle -\cos\left(a\right)$

ens={0,pi/6,pi/4,pi/3,pi/2,pi}
map(cos,ens)

[$\displaystyle 1$, $\displaystyle \frac{1}{2}$, $\displaystyle -1$, $\displaystyle 0$, $\displaystyle \frac{1}{2} \, \sqrt{3}$, $\displaystyle \frac{1}{2} \, \sqrt{2}$]

map(sin,ens)

[$\displaystyle 0$, $\displaystyle \frac{1}{2} \, \sqrt{3}$, $\displaystyle 0$, $\displaystyle 1$, $\displaystyle \frac{1}{2}$, $\displaystyle \frac{1}{2} \, \sqrt{2}$]

map(tan,ens)

[$\displaystyle 0$, $\displaystyle \sqrt{3}$, $\displaystyle 0$, $\displaystyle \infty$, $\displaystyle \frac{1}{3} \, \sqrt{3}$, $\displaystyle 1$]

#Il y a bcp de valeurs communes entre les résultats des sinus et des cosinus par contre pour les valeurs de tangente c'est différent ! en clair on a la division des valeurs des sinus par les valeurs de cosinus
line([(0,0), (1,2), (1/2,pi), (1/2,pi/2)], color='darkgreen', thickness=3)

var('x')
var('y')
a=1
b=6
r=2
circle((a,b),r)

x y

var('t')
parametric_plot([a+r*cos(t),b+r*sin(t)],-10,10, color='green', thickness=3, fill = True)

t