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By Kenneth Kuttler

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For example, in the following picture, the two triangles are similar because the angles are the same. ✑ ✑ ✑ B ✑ ✑ c✑ a ✑ ✑ ✑ c∗ ✑ ✑ ✑ ✑ A✑ b B a∗ ✑ C ✑ ✑ A ✑ b∗ C The fundamental axiom for similar triangles is the following. 2 If two triangles are similar then the ratios of corresponding parts are the same. 3 Two lines in the plane are said to be parallel if no matter how far they are extended, they never intersect. 4 If two lines l1 and l2 are parallel and if they are intersected by a line, l3 , the alternate interior angles are shown in the following picture labeled as α.

11. Prove by induction that n < 2n for all natural numbers, n ≥ 1. 44 THE REAL NUMBERS 12. Prove by the binomial theorem and Problem 9 that the number of subsets of a given finite set containing n elements is 2n . 13. Let n be a natural number and let k1 + k2 + · · ·kr = n where ki is a non negative integer. The symbol n k1 k2 · · · kr denotes the number of ways of selecting r subsets of {1, · · ·, n} which contain k1 , k2 ···kr elements in them. Find a formula for this number. n 14. Is it ever the case that (a + b) = an + bn for a and b positive real numbers?

4 Let x, y ∈ R. Then rx+y = rx + ry + 2kπ for some k ∈ Z. Proof: By definition, x + y = 2πp + rx+y , x = 2πp1 + rx , y = 2πp2 + ry . 6. THE TRIGONOMETRIC FUNCTIONS 61 From this the result follows because ≡−k 0 = ((x + y) − x) − y = 2π ((p − p1 ) − p2 ) + rx+y − (rx + ry ) . Let z ∈ R and let p (z) denote the point on the unit circle determined by the length rz whose coordinates are cos z and sin z. Thus, starting at (1, 0) and moving counter clockwise a distance of rz on the unit circle yields p (z) .

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Calculus:Theory And Applications by Kenneth Kuttler


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