Unformatted text preview: . In graphics programming, the following standard trigonometric identities are often used to rotate objects.
cos(x + y ) = cos(x) cos(y ) − sin(x) sin(y )
sin(x + y ) = sin(x) cos(y ) + cos(x) sin(y ) Use these formulas to prove that ∀n ∈ Z+ , ∀x ∈ R,
(cos(x) + i sin(x))n = cos(nx) + i sin(nx).
Recall that i is the “imaginary unit”, i.e., the square root of −1 (so i2 = −1). Do not convert this problem to exponential notation and try to solve it in that form!
Hint: This is actually simpler than it looks. If your equations seem very complicated, it is most likely
because of a mistake in your algebra — everything is supposed to simplify nicely.
2. Deﬁne a function f recursively, as follows:
f (1) = 1,
f (2) = 5,
f (n) = 5f (n − 1) − 6f (n − 2) ∀ n 3. Use complete induction to prove that f (n) = 3n − 2n for every positive integer n.
3. Recall the recursive deﬁnition of complete binary tree s:
• a single node is a complete binary tree, • if T1 and T2 are complete binary trees with the same height, then the tree constructed
by placing T1 and T2 under a new root node (as illustrated below on the left) is also a
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