how would you represent the area outside given y= 4cos(2theta) but outside y=4 ?
4 cos anything is never bigger than 4 so there is no overlap here
Will u please answer it with detailed
look at the graphs here:
http://www.wolframalpha.com/input/?i=plot+y+%3D+4cos%282x%29%2C+y%3D4
Now you have to clarify what you mean by "outside" a line
To represent the area outside the region bounded by the curves y = 4cos(2theta) and y = 4, we can use polar coordinates to define the boundaries of the region.
First, let's understand the curves. The equation y = 4cos(2theta) represents a cardioid, and the equation y = 4 represents a circle centered at the origin with a radius of 4.
To visualize the region, we need to plot these curves on a polar coordinate system. This can be achieved by using the equation r = f(theta), where r represents the distance from the origin, and theta represents the angle.
The curve y = 4cos(2theta) can be rewritten in polar coordinates as r = 4cos(2theta), and the circle y = 4 can be rewritten as r = 4.
To determine the region outside the circle but inside the cardioid, we need to find the boundaries for theta.
1. Start by setting the two equations equal to each other and solve for theta:
4cos(2theta) = 4
2. Simplify the equation:
cos(2theta) = 1
3. Since the range of the cosine function is -1 to 1, we need to find the values of theta that satisfy cos(2theta) = 1.
When cos(2theta) = 1, it means that 2theta is an even multiple of pi.
So, 2theta = 0, 2pi, 4pi, 6pi, ...
Solving for theta, we get:
theta = 0, pi, 2pi, 3pi, ...
4. Now we have the values of theta that define the boundaries of the shaded region.
Since theta ranges from 0 to 2pi to cover a complete circle, the region outside the given cardioid but inside the circle can be defined as:
0 <= theta < pi, pi <= theta < 2pi
This represents the portion of the polar coordinate system that lies outside the cardioid but inside the circle y = 4.