Find the polar equation r=8cos3theta, find the maximum value of lrl and any zeros of r. Verify your answers numerically.

well,

|cos3θ| <= 1

cos3θ=0 when 3θ is an odd multiple of π/2

To find the maximum value of |r| and any zeros of r, we need to analyze the given polar equation r = 8cos(3θ).

1. Maximum value of |r|:
We know that |cos(θ)| has a maximum value of 1. Therefore, |r| will be maximum when cos(3θ) = 1.

To find the maximum value of |r|, we solve for θ when cos(3θ) = 1:
cos(3θ) = 1
3θ = 0 (taking the inverse cosine function)
θ = 0

So, the maximum value of |r| occurs when θ = 0.

Substituting θ = 0 into r = 8cos(3θ), we find |r|:
r = 8cos(3(0))
= 8cos(0)
= 8(1)
= 8

Therefore, the maximum value of |r| is 8.

2. Zeros of r:
Zeros of r occur when cos(3θ) = 0.

To find the zeros of r, we solve for θ when cos(3θ) = 0:
cos(3θ) = 0

The solutions for cos(3θ) = 0 are:
3θ = π/2 + nπ/3 (where n is an integer)
θ = (π/2 + nπ/3) / 3

Therefore, the zeros of r occur at θ = π/6, π/2, and 5π/6.

To verify these results numerically, we can evaluate r for different values of θ and check if the maximum value and zeros match the values obtained analytically:

For θ = 0, r = 8cos(3(0)) = 8(1) = 8 (maximum value)

For θ = π/6, r = 8cos(3(π/6)) = 8cos(π/2) = 8(0) = 0 (zero)

For θ = π/2, r = 8cos(3(π/2)) = 8cos(3/2π) = 8(-1) = -8 (zero)

For θ = 5π/6, r = 8cos(3(5π/6)) = 8cos(5π/2) = 8(0) = 0 (zero)

As we can see, the numerical verification confirms that the maximum value of |r| is 8, and the zeros of r occur at θ = π/6, π/2, and 5π/6.

To find the maximum value of lrl, we need to find the maximum value of r. Recall that in polar coordinates, r represents the distance from the origin to a point (r, θ) in the polar plane.

Let's start by analyzing the given polar equation r = 8cos(3θ) in terms of its components:

1. The "r" in the equation represents the radial distance from the origin.
2. The "θ" in the equation represents the angle made with the positive x-axis.

To find the maximum value of r, we can look at the range of values that cos(3θ) can take on. The cosine function has a maximum value of 1, occurring when the angle inside the cosine function is 0.

So, to find the maximum value of r, we need to find the angle θ that makes cos(3θ) equal to 1:

cos(3θ) = 1

Solving this equation for θ, we have:

3θ = 0

θ = 0

Therefore, to find the maximum value of r, we substitute θ = 0 into the polar equation:

r = 8cos(3(0))
r = 8cos(0)
r = 8(1)
r = 8

Hence, the maximum value of lrl or r is 8.

Now, let's find any zeros of r. Zeros occur when r equals zero. To find these values, we set the equation r = 8cos(3θ) equal to zero and solve for θ:

8cos(3θ) = 0

cos(3θ) = 0

To find the values of θ that make the cosine function equal to zero, we look for the angles where the cosine function crosses the x-axis. These angles occur at intervals of π radians.

Therefore, we solve for θ by setting 3θ equal to π/2, π, 3π/2:

3θ = π/2, π, 3π/2

θ = π/6, π/3, π/2

So, the zeros of r occur at θ = π/6, π/3, and π/2.

To verify our answers numerically, we can substitute these values of θ into the polar equation r = 8cos(3θ) and check if the resulting values of r are indeed zero or 8 for the maximum:

For θ = π/6:
r = 8cos(3(π/6)) = 8cos(π/2) = 8(0) = 0

For θ = π/3:
r = 8cos(3(π/3)) = 8cos(π) = 8(-1) = -8

For θ = π/2:
r = 8cos(3(π/2)) = 8cos(3π/2) = 8(0) = 0

Therefore, we have verified numerically that the maximum value of r is 8 and the zeros of r occur at θ = π/6 and θ = π/2.