Final answer:
Sin(x) and e^x are linearly independent because their Wronskian is never zero; sin(x) and cos(x) are not multiples of each other, and e^x is never zero.
Step-by-step explanation:
To determine if the functions sin(x) and ex are linearly independent, we can employ the Wronskian test.
Two functions f and g are linearly independent if their Wronskian, W(f, g) = f(x)g'(x) - g(x)f'(x), is nonzero for at least one point in the interval being considered.
To apply the Wronskian to sin(x) and ex, we calculate:
f(x) = sin(x)
g(x) = ex
f'(x) = cos(x)
g'(x) = ex
W(sin(x), ex) = sin(x) x (ex) - ex x cos(x)
Since ex is never zero and sin(x) and cos(x) are not multiples of each other, the Wronskian is never zero. Therefore, sin(x) and ex are linearly independent functions.