what are the horizontal and vertical intercepts :r=4-4cos(theta) please show work
Where are you getting stuck on this problem? Have you done any work on it yet?
horizontal intercepts lie on the lines where theta = 0 or pi, right? (The x-axis)
vertical intercepts lie on the y-axis, where theta = ...
To find the horizontal and vertical intercepts of the equation r = 4 - 4cos(theta), we need to convert the equation from polar coordinates to rectangular coordinates.
First, we can rewrite the equation as:
x = r * cos(theta)
y = r * sin(theta)
Substituting the given equation, we have:
x = (4 - 4cos(theta)) * cos(theta)
y = (4 - 4cos(theta)) * sin(theta)
Now, let's find the horizontal intercepts, which occur when y = 0:
(4 - 4cos(theta)) * sin(theta) = 0
To solve this equation, we have two possibilities:
1. sin(theta) = 0:
In this case, theta can be any multiple of pi (π), which means the horizontal intercepts occur when theta = 0 or theta = pi.
2. 4 - 4cos(theta) = 0:
Solving for theta gives:
cos(theta) = 1
theta = 0
Therefore, the horizontal intercept occurs at theta = 0.
Next, let's find the vertical intercepts, which occur when x = 0:
(4 - 4cos(theta)) * cos(theta) = 0
Again, we have two possibilities:
1. cos(theta) = 0:
In this case, theta can be any odd multiple of pi/2 (π/2), which means the vertical intercepts occur when theta = pi/2 or theta = 3pi/2.
2. 4 - 4cos(theta) = 0:
Solving for theta gives:
cos(theta) = 1
theta = 0
Therefore, the vertical intercept occurs at theta = 0.
To summarize, the horizontal intercept occurs at theta = 0 and the vertical intercepts occur at theta = 0, pi/2, and 3pi/2.