Tiêu đề 37 Trigonometric Functions.pdf ✅

MSTC 37: Trigonometric Functions 1. Six Basic Trigonometric Functions The trigonometric ratios from the previous discussion can also be used as functions. As such, their functions will have domains and ranges. When used as functions, all angles are defaulted to radian units. Consider the pair of functions y = sin x y = cos x The domain of these functions is the set of real numbers since any angle has both sine and cosine. The range of these functions is just between −1 and 1. For the pair of functions y = tan x y = sec x The domain of these functions is the set of real numbers except coterminal angles of π 2 and − π 2 . The range of these functions is the set of real numbers. For the pair of functions y = cot x y = csc x The domain of these functions is the set of real numbers except coterminal angles of 0 and π. The range of these functions is the set of real numbers. 2. Archaic Trigonometric Functions Consider a unit circle (a circle with a radius of 1 unit) centered at the origin, From the figure, the length of AC and BC are the cosine and the sine of the angle, respectively, since AC is the side adjacent to θ and BC is the side opposite to θ. AC = BD = cos θ BC = AD = sin θ
Here, the other trigonometric functions represent the following lengths: AE = csc θ AF = sec θ BE = cot θ BF = tan θ Historically, some of the lengths represented in the figures have their trigonometric functions related to them. They are: DG = cvs θ = 1 − sin θ GE = excsc θ = csc θ − 1 CH = vers θ = 1 − cos θ HF = exsec θ = sec θ − 1 Another set of archaic (“archaic” means old) functions are those prefixed with “ha-”, which is just half of the function indicated. These notations are not used that much anymore and are instead directly expressed with their equivalents. These functions are commonly used in surveying, particularly in routes. 3. Inverse Trigonometric Functions An inverse trigonometric function is a function that undoes a trigonometric function. They are indicated by the prefix “arc-” continued by the trigonometric function they invert. When about principal values (in between 0 to 2π), the prefix “Arc-” is used (the A is in uppercase). The domain of an inverse function is the range of the original, and the range of an inverse function is the domain of the original. Inverse Trigonometric Functions. y = sin x ⇒ x = arcsin y y = cos x ⇒ x = arccos y y = tan x ⇒ x = arctan y y = csc x ⇒ x = arccsc y y = sec x ⇒ x = arcsec y y = cot x ⇒ x = arccot y 4. Hyperbolic and Inverse Hyperbolic Functions Hyperbolic functions are exponential functions that are analogous to trigonometric functions. The six hyperbolic functions are the following: Hyperbolic Functions. sinh x = e x − e −x 2 cosh x = e x + e −x 2 tanh x = e x − e −x e x + e −x