# Calculus

The logarithm to the base e, is called the natural logarithm and its properties are developed in a later chapter. The natural logarithm is given a special notation. Whenever the base b = e one can write either y = loge x = ln x or y = log x (1.23) That is, if the notation ln is used or whenever the base is not specified in using logarithms, it is to be understood that the base b = e is being employed. In this special case one can show y = ex = exp(x) ⇐⇒ x = ln y (1.24) which gives the identities ln(ex) = x, x ∈ R and eln x = x, x > 0

The Trigonometric Functions The ratio of sides of a right triangle are used to define the six trigonometric functions associated with one of the acute angles of a right triangle. These definitions can then be extended to apply to positive and negative angles associated with a point moving on a unit circle. The six trigonometric functions associated with a right triangle are sine cosine tangent cotangent secant cosecant which are abbreviated respectively as sin, tan, sec, cos, cot, and csc . Let θ and ψ denote complementary angles in a right triangle as illustrated above. The six trigonometric functions associated with the angle θ are sin θ = y r = opposite side hypotenuse , cos θ = x r = adjacent side hypotenuse , tan θ = y x = opposite side adjacent side, cot θ = x y = adjacent side opposite side, sec θ = r x = hypotenuse adjacent side csc θ = r y = hypotenuse opposite side