Answer:r=2 cos^4(theta)
Step-by-step explanation:To find the values of theta where the maximum r-values occur on the graph of the polar equation r = 2 cos^4(theta), we need to find where the derivative of r with respect to theta is equal to zero, since the maximum r-values occur at these points.
First, we can simplify the equation by using the identity cos(2theta) = 2cos^2(theta) - 1 and substituting cos^2(theta) = (1 + cos(2theta))/2. This gives:
r = 2 cos^4(theta) = 2(1/2 + 1/2 cos(2theta))^2 = 1 + cos(2theta) + cos^2(2theta)/2.
Next, we can take the derivative of r with respect to theta, using the chain rule:
dr/dtheta = -sin(2theta) - 2cos(2theta)sin(2theta).
Setting this equal to zero and factoring out sin(2theta), we get:
sin(2theta)(-1 - 2cos(2theta)) = 0.
This equation is satisfied when sin(2theta) = 0 or cos(2theta) = -1/2.
When sin(2theta) = 0, we have 2theta = k*pi for some integer k. Therefore, theta = k*pi/2.
When cos(2theta) = -1/2, we have 2theta = 2*pi/3 + 2*k*pi or 2theta = 4*pi/3 + 2*k*pi for some integer k. Therefore, theta = pi/3 + k*pi or theta = 2*pi/3 + k*pi.
These are the values of theta where the maximum r-values occur on the graph of the polar equation r = 2 cos^4(theta).