Let G be the graph of the parametric equations x=cos(4t)and y=sin(6t). What is the length of the smallest interval I such that the graph of these equations for all t in I produces the entire graph G?
jwM
To find the length of the smallest interval I that produces the entire graph G, we need to determine the range of values for t that cover the full period of the trigonometric functions used (cos(4t) and sin(6t)).
For the x-coordinate, x = cos(4t), the period of the cosine function is 2π divided by the coefficient of t, which is 4. So, the period of the x-coordinate function is 2π/4 = π/2.
For the y-coordinate, y = sin(6t), the period of the sine function is 2π divided by the coefficient of t, which is 6. So, the period of the y-coordinate function is 2π/6 = π/3.
To find the smallest interval I that produces the entire graph, we need to find the least common multiple (LCM) of the two periods, which is the smallest interval that covers both periods simultaneously.
To find the LCM of π/2 and π/3, we convert both to a common denominator:
LCM (π/2, π/3) = LCM (3π/6, 2π/6) = LCM (3π, 2π)/6.
Now, since the LCM of 3π and 2π is 6π, the smallest interval I is 6π/6, which simplifies to just π.
Therefore, the length of the smallest interval I such that the graph of the parametric equations x = cos(4t) and y = sin(6t) produces the entire graph G is π.