Mathematical Formulas for Industrial and Mechanical Engineering

Arc Cosine

Let function g: [0, π] ! [21, 1], where g(x) 5 cos x . Therefore, its inverse func-

tion is defined by g

21

:[21, 1] ! [0, π], where g

21

(x) 5 cos

21

x and is called the

arc cosine function.

Also, y 5 cos

21

x 3 x 5 cos y.

Arc Tangent

Let h:(2π/2, π/2) ! ℜ,whereh(x) 5 tan x. Therefore, its inverse function is

defined by h

21

: ℜ ! (2π/2, π/2), where h

21

(x) 5 tan

21

x and is called the arc tan-

gent function.

Hence, if y 5 tan

21

x 3 tan y 5 x.

4.9 Solutions of Trigonometric Equations

For tan θ 5 k, the general solution is θ 5 nπ 1 α, nAΖ, α 5 tan

21

k.

For cos θ 5 k, where jkj# 1, the general solution is θ 5 2nπ 6 α, nAΖ,

α 5 cos

21

k.

For sin θ 5 k, where jk