Page 43 - Standard Handbook Of Petroleum & Natural Gas Engineering

32 Mathematics

(text continued from page 29)
Graphs of Trigonometric Functions

Graphs of the sine and cosine functions are identical in shape and periodic
with a period of 360". The sine function graph translated f90" along the x-axis
produces the graph of the cosine function. The graph of the tangent function
is discontinuous when the value of tan 9 is undefined, that is, at odd multiples
of 90" (. . ., go", 270", . . .). For abbreviated graphs of the sine, cosine, and
tangent functions, see Figure 1-29.
Inverse Trigonometric Functions

The inverse sine of x (also referred to as the arc sine of x), denoted by sin-'x,
is the principal angle whose sine is x, that is,

y = sin-'x means sin y = x
Inverse functions COS-'^ and tan-'x also exist for the cosine of y and the tangent
of y. The principal angle for sin-'x and tan-'x is an angle a, where -90" < a <
go", and for cos-'x, O"< a < 180".
Solution of Plane Triangles
The solution of any part of a plane triangle is determined in general by any
other three parts given by one of the following groups, where S is the length
of a side and A is the degree measure of an angle:
AAS
SAS
sss
The fourth group, two sides and the angle opposite one of them, is ambiguous
since it may give zero, one, or two solutions. Given an example triangle with
sides a, b, and c and angles A, B, and C (A being opposite a, etc., and A + B
+ C = 180"), the fundamental laws relating to the solution of triangles are
1. Law of Sines: a/(sin A) = b/(sin B) = c/(sin C)
.
.
2. Law of Cosines': 2 = ai + b2 1 2ab cos C

4*i" x

Figure 1-29. Graphs of the trigonometric functions.