Repeat Exercise 45 for the functions f(t) = 3 sin t - 4 sin³ t, g(t) = 1 - 8 sin² t + 8 sin⁴ t, h(t) = 5 sin t - 20 sin³ t + 16 sin⁵ t, in the vector space Span {I, sin t, sin² t, ..., sin⁵ t}.

Data from in Exercise 45

The vector space H = Span {1, cos² t, cos⁴ t, cos⁶ t} contains at least two interesting functions that will be used in a later exercise:

Study the graph of f for 0 ≤ t ≤ 2π, and guess a simple formula for f(t). Verify your conjecture by graphing the difference between 1 + f(t) and your formula for f(t). (Hope- fully, you will see the constant function 1.) Repeat for g.

f(t) = 1-8 cos t + 8 cost g(t) = 1 + 18 cos t - 48 cos4 t + 32 cost