Exploring powers of sine and cosine a. Graph the functions
Chapter 7, Problem 73(choose chapter or problem)
Exploring powers of sine and cosine a. Graph the functions f11x2 = sin2 x and f21x2 = sin2 2x on the interval 30, p4. Find the area under these curves on 30, p4. b. Graph a few more of the functions fn 1x2 = sin2 nx on the interval 30, p4, where n is a positive integer. Find the area under these curves on 30, p4. Comment on your observations. c. Prove that 1 p 0 sin2 1nx2 dx has the same value for all positive integers n. d. Does the conclusion of part (c) hold if sine is replaced with cosine? e. Repeat parts (a), (b), and (c) with sin2 x replaced with sin4 x. Comment on your observations. f. Challenge problem: Show that for m = 1, 2, 3, c, L p 0 sin2m x dx = L p 0 cos2m x dx = p # 1 # 3 # 5 g12m - 12 2 # 4 # 6 g2m .
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