Parametric Equations
Name:
Goal: To graph the parametric equations ( x(t) = cos(t), y(t) = sin(t)) for t in [0 , 2 Pi].
1. Fill in the following table. Be sure to calculate the values of sin and cos using radians.
t
0
Pi/6
Pi/4
Pi/3
Pi/2
3Pi/4
Pi
5 Pi/4
3 Pi/2
7 Pi/4
2 Pi
x(t) = cos (t)
y(t) = sin(t)
a. Graph the sin(t) function. More precisely, make a graph of (t, x(t)) = ( t, sin(t))
b. Graph the cos(t) function. More precisely, make a graph of (t, y(t)) = ( t, cos(t))
More questions on other side.
c. Make a graph of (x(t), y(t)). By this, I mean for each t value in your table, graph the
pair of points (x(t), y(t)) on the xy plane. On your picture, label the xy value and also
label the t value. When you have plotted all the points from the table, draw the
continuous curve that connects the points – this is the curve you would get if you made a
table with many more t values. Finally put an arrow on the curve indicating the direction
of motion: i.e. as t increases, which direction do you move along the curve?
d. Show that the curve generated by the parametric equation ( x(t) = cos(t), y(t) = sin(t))
satisfies the equation:
x2 + y 2 = 1
Hint: What do x(t) and y(t) equal?
Hence what special curve does the parametric equation ( x(t) = cos(t), y(t) = sin(t))
produce?
e. Given a parametric curve (x(t), y(t)), the tangent vector to the curve is the vector
(x’(t) , y’(t)). What is the general expression for the tangent vector to the curve
( x(t) = cos(t), y(t) = sin(t))?
f. At t = Pi/2, what is the tangent vector to the curve?
This tangent vector must get placed at the appropriate spot – at the point (x(Pi/2),
y(Pi/2)). What is the point (x(Pi/2), y(Pi/2))?
In your drawing in part c, attach this tangent vector at the appropriate point.
g. At t = Pi/4, what is the tangent vector to the curve?
This tangent vector must get placed at the appropriate spot – at the point (x(Pi/4),
y(Pi/4)). What is the point (x(Pi/4), y(Pi/4))?
In your drawing in part c, attach this tangent vector at the appropriate point.
h. Using parametric equations we can give a more sophisticated
way to write the equation

of a line. Suppose we want the line through the point P = (x 0 , y 0 ) in the direction of the

vector V = (v x ,v y ) . Then the line is given by the formula


l(t) = (x(t), y(t)) = P €
+ tV = (x 0 , y 0 ) + t(v x ,v y )
€
We can split the formulas into two parts:
x(t) = x 0 + tv x
€
.
y(t) = y 0 + tv y
Using the parametric equations for a line,

h.i. Give the equation of the€tangent
line at the point P = (x(Pi/2), y(Pi/2)) in the

direction of the tangent vector V = (x’(Pi/2), y’(Pi/2)). Then draw this line on your
picture.
€
€

P
= (x(Pi/4), y(Pi/4)) in the
h.ii. Give the equation of the tangent
line
at
the
point

direction of the tangent vector V = (x’(Pi/4), y’(Pi/4)). Then draw this line on your
picture.
€
€
2. Repeat questions (a-h) but for the parametric equations (x(t) = t cos(t), y(t) = t sin(t)).
For (d), calculate the value of x2 + y 2. What does this value for x2 + y 2 tell us about the
curve given by (x(t) = t cos(t), y(t) = t sin(t))?
In filling in the table, calculate the following values.
t
0
Pi/4
Pi/2
3Pi/4
Pi
5 Pi/4
3 Pi/2
7 Pi/4
2 Pi
2 Pi + Pi/2
3 Pi
3 Pi + Pi/2
4 Pi
x(t) = t * cos (t)
y(t) = t * sin(t)