Find an interval for theta where the graph is traced only once:
r = 2-8cos(theta)
Please help this is confusing.
Is it just 0≤θ≤2pi?
yes that is one period
To find the interval for theta where the graph of the equation r = 2-8cos(theta) is traced only once, we need to determine the values of theta that result in a unique, non-overlapping graph.
First, let's understand the equation. The equation r = 2-8cos(theta) represents a polar equation in the form of r = a + bcos(theta), where 'a' and 'b' are constants. The variable 'r' represents the distance from the origin to a point in the polar coordinate system, and 'theta' represents the angle the line connecting the origin and the point makes with the positive x-axis.
To determine the interval where the graph is traced only once, we need to find the range of values for theta that covers one complete revolution, without any repetition or overlap.
In this case, the equation r = 2-8cos(theta) has a constant term '2' and a cosine function with a coefficient of '8'.
The cosine function has a period of 2π, meaning that it repeats every 2π radians. However, the coefficient '8' in front of the cosine function stretches or compresses the graph along the x-axis.
Since the coefficient is positive (8 > 0), the graph will be compressed horizontally, resulting in a shorter period.
To determine the compressed period, we can divide the normal period (2π) by the absolute value of the coefficient (8).
Compressed period = 2π / |8| = π/4
So, the graph of r = 2-8cos(theta) will complete one full revolution in the interval of theta = 0 to theta = 2π, but within this interval, there will be multiple repetitions of the graph.
To find the interval where the graph is traced only once, we need to consider the range of theta values that corresponds to the compressed period.
Since the compressed period is π/4, the graph will complete four revolutions in the interval of theta = 0 to theta = 2π. Therefore, to ensure the graph is traced only once, we need to restrict theta to a smaller interval.
To find this interval, we can divide the interval theta = 0 to theta = 2π into four equal parts:
theta = [0, π/4), [π/4, π/2), [π/2, 3π/4), [3π/4, π)
Thus, the interval [0, π/4) is the range of theta values where the graph of r = 2-8cos(theta) is traced only once.