show that a function f(x)=sin(x) and f(x)=cos(x) are continuous at any point of the interval [0,2 pi]

Question

real analysishI6s

Answer 1

A circle of radius 1 has a continuous circumference. If the circle is drawn on Cartesian orthogonal axes, which we can call Y and X, then any point P(X,Y) on the circumference can be described in terms of the angle the radius makes with the X-axis. We can call the angle x. The X-coordinate of P is cosx and the Y-coordinate is sinx, so the point P is (cosx,sinx). As x goes from 0 to 2π, P describes the circumference of the circle, which is continuous. Therefore cosx and sinx must be continuous functions.