**Polar Curves**

**Q:- Plot the curve :r = 1+cos(t)**

Ans: Give the command at sage prompt.

t,s=var(‘t,s’)

s=polar_plot(1+cos(t),(t,0,pi),aspect_ratio = 1, color = ‘red’)

s.show()

s.save(‘cardoid.png’)

Note: Aspect_ratio=1 forces the x and y axes to use the same scale. In other words, one unit on the x-axis takes up the same number of pixels on the screen as one unit on y-axis.

s.save(‘cardoid.png’) will save the image formed as output of above curve as cardoid.png in home folder.

**Q:- r=cos(2t)**

Ans:

t,s=var(‘t,s’)

s=polar_plot(cos(2*t),(t,0,pi),aspect_ratio = 1, color = ‘red’)

s.show()

s.save(‘cardoid.png’)

**Parametric Equations**

**Q:- x=cos(x) , y= sin(x)**

Ans:

x = var(‘x’)

pp = parametric_plot((cos(x),sin(x)),(x,0,2*pi),fill = True, fillcolor =’purple’)

pp.show(aspect_ratio =1, figsize =(3,3), frame= True)

**Q:-x= t+sin(t) , y= 1+cos(t)**

var(‘t’)

f(t) =( t+sin(t))

s(t)= 1+cos(t)

pp= parametric_plot((f(t),s(t)),(t,-2*pi, 2*pi),fill = True, fillcolor =’blue’)

pp.show()

**Simple Equations**

**Q:- plot sine and cosine waves.**

Ans:

p1=plot(sin,(-2*pi,2*pi), thickness = 2.0, rgbcolor = (0.5,1,0), legend_label=’sin(x)’)

p2=plot(cos,(-2*pi,2*pi), thickness = 3.0, color =’purple’,alpha= 0.5, legend_label=’cos(x)’)

plt = p1+p2

plt.axes_labels([‘x’,’f(x)’])

show(plt)

**Q:- x^2+ y^2 -4**

Ans:-

t,x,y =var(‘t,x,y’)

t= plot3d(x^2+y^2-4,(x,-2,2),(y,-2,2))

t.save(‘raj.png’)

t.show()

**Q:- Plot y=x^2**

Ans:-

f(x) = x^2

p=plot(f(x),(x,-pi/2,pi/2),color=’blue’)

p.show()