Identify each polar equation by changing the equation to rectangular coordinates.
Let t = theta
(1) t = -pi/4
(2) r(sin(t)) = -2
To identify each polar equation by changing them to rectangular coordinates, we will use the following relationships:
x = r*cos(t)
y = r*sin(t)
Let's start with equation (1):
(1) t = -pi/4
To convert this equation to rectangular coordinates, we substitute x and y for r*cos(t) and r*sin(t), respectively:
t = atan2(y, x)
Now we can solve for x and y by substituting t = -pi/4:
x = r*cos(-pi/4)
x = r/sqrt(2)
y = r*sin(-pi/4)
y = -r/sqrt(2)
Therefore, the rectangular coordinates of equation (1) are:
x = r/sqrt(2)
y = -r/sqrt(2)
Now let's move on to equation (2):
(2) r*sin(t) = -2
To convert this equation to rectangular coordinates, we use the same x and y substitutions:
r*sin(t) = y
Substituting y for -2:
r*sin(t) = -2
Now we can solve for x by using the relationship x = r*cos(t):
x = r*cos(t)
Therefore, the rectangular coordinates of equation (2) are:
x = r*cos(t)
y = -2
To summarize:
Equation (1) in rectangular coordinates is:
x = r/sqrt(2)
y = -r/sqrt(2)
Equation (2) in rectangular coordinates is:
x = r*cos(t)
y = -2