Let G be the graph of the parametric equations
x = cos(4t),
y = sin(6t).
What is the length of the smallest interval P such that the graph of these equations for all t E P produces the entire graph G?
(The "E" represents the symbol for "is in" that looks kind of like a rounded E)
does this help?
https://www.wolframalpha.com/input/?i=plot+x+%3D++cos(4t),+y+%3D+sin(6t)
So would the interval be (-inf, 1] ??
But that wouldn't be the right answer since it's asking for a length.
cos(4t) has period pi/2
sin(6t) has period pi/3
so, you get the entire curve using the interval
[0,pi]
as indicated by the note next to the graph at the suggested URL.
To find the length of the smallest interval P such that the graph of the parametric equations produces the entire graph G, we need to determine the period of each equation separately.
1. Period of x = cos(4t):
The period of the cosine function, cos(t), is 2π. However, in this case, we have cos(4t), which means the argument inside the cosine function is multiplied by 4. As a result, the period of x = cos(4t) is 2π/4 = π/2.
2. Period of y = sin(6t):
Similar to the cosine function, the period of the sine function, sin(t), is also 2π. Here, we have sin(6t), so the argument inside the sine function is multiplied by 6. Consequently, the period of y = sin(6t) is 2π/6 = π/3.
To determine the smallest interval P that produces the entire graph G, we find the least common multiple (LCM) of the periods of x and y. The LCM of π/2 and π/3 can be calculated as follows:
LCM(π/2, π/3) = (2π/3) / (GCD(2π/3, π/2)),
where GCD represents the greatest common divisor. In this case, GCD(2π/3, π/2) is equal to π/6.
LCM(π/2, π/3) = (2π/3) / (π/6) = 4π/3.
Therefore, the length of the smallest interval P is 4π/3.