6: Homework The Nature of Sin
Denition 1 =2
1
1 x2 dx.
1
1. Geometrically motivate this denition (with picture(s)!). Denition 2 For 1 x 1, x 1 x2 + A(x) = 2
1
1 t2 dt.
x
2. Geometrically motivate this denition. Hint: the rst term is the area of a triangle. 3. Calculate A (x) and A(1), A(1). Denition 3 The function cos : [0, ] R is dened by letting cos x be the unique number in [1, 1] such that A(cos x) = x . The function sin : [0, ] R 2 is dened by: sin x = 1 (cos x)2 . 4. Prove that cos and sin are well dened by showing that A is a bijection from [1, 1] to [0, ]. 5. Consider a line in the x, y plane through the origin that makes an angle of with the positive x-axis, where 0 . Prove that the line intersects the unit circle (i.e. the circle centered at the origin with radius 1) at the point (cos , sin ). Denition cos, 4 sin : R R are dened as the unique functions agreeing with Denition 3 on [0, ] that satisfy: sin x = sin(x) cos x = cos(x) sin(x + 2 ) = sin x cos(x + 2 ) = cos x.
1
6. Prove that: sin x = cos x, 7. Dene f (x) = Prove that f is continuous at 0. Recall that a function f : R R is twice-dierentiable if f is dierentiable and f is dierentiable. Naturally, we write f for the derivative of f . 8. Suppose that f : R R is twice-dierentiable and satises: f +f =0 f (0) = 0 f (0) = 0. Prove that f ...