Final answer:
To find h⁴(π/3), we substitute the value of π/3 into h(x), use the given value of g(π/3), and calculate the cosine function. We then raise the result of h(π/3) to the fourth power to obtain h⁴(π/3) ≈ 16, assuming the cosine of a number very close to zero is approximately equal to 1.
Step-by-step explanation:
We are tasked with finding the value of h⁴(π/3), given that h(x)=g(x) \times cos(1/(2x)) and we know the values of g(π/3) = 2 and g²(3) = 4. The notation g²{3} seems to be a typographical error or irrelevant information, so we will focus on the relevant given values.
To find h⁴(π/3), we first substitute the value of x with π/3 into the function h:
h(π/3) = g(π/3) \times cos(1/(2 \times π/3))
Since we are given g(π/3) = 2, we can substitute this value to obtain:
h(π/3) = 2 \times cos(1/(2 \times π/3))
h(π/3) = 2 \times cos(3/(2π))
Now, we need to find cos(3/(2π)), which is not explicitly given but assuming the standard trigonometric values, we find cos(3/(2π)) ≈ 1 (as 3/(2π) approaches zero).
Therefore, h(π/3) ≈ 2 \times 1 = 2.
Finally, we raise this result to the fourth power to find h⁴(π/3):
h⁴(π/3) ≈ 2⁴ = 16