Final answer:
The fundamental period of the function f1(x) = sin7x cos14x is found using a trigonometric identity to simplify the expression and calculate the individual periods of the resulting sinusoidal functions. The fundamental period is the least common multiple of these individual periods, which results to π/7.
Step-by-step explanation:
To find the fundamental period of the function f1(x) = sin7x cos14x, we first need to use a trigonometric identity to simplify the expression. The identity sin A cos B = ½(sin(A + B) + sin(A - B)) allows us to write the function as:
f1(x) = ½(sin(7x + 14x) + sin(7x - 14x)) = ½(sin(21x) + sin(-7x)) = ½(sin(21x) - sin(7x))
Now, we have two sinusoidal functions sin(21x) and sin(7x). The fundamental period of a sinusoidal function sin(kx) is π/k. Therefore, the periods for sin(21x) and sin(7x) are:
Period of sin(21x) = π/21
Period of sin(7x) = π/7
To find the fundamental period of the combination, we need the least common multiple (LCM) of these two periods. The LCM of π/21 and π/7 is π/7, which is the fundamental period of the original function f1(x).
Therefore, the fundamental period of f1(x) = sin7x cos14x is π/7.