4 Trigonometry and Complex Numbers

[Pages:42]4 Trigonometry and Complex

Numbers

Trigonometry developed from the study of triangles, particularly right triangles, and the relations between the lengths of their sides and the sizes of their angles. The trigonometric functions that measure the relationships between the sides of similar triangles have far-reaching applications that extend far beyond their use in the study of triangles. Complex numbers were developed, in part, because they complete, in a useful and elegant fashion, the study of the solutions of polynomial equations. Complex numbers are useful not only in mathematics, but in the other sciences as well.

Trigonometry

Most of the trigonometric computations in this chapter use six basic trigonometric func-

tions The two fundamental trigonometric functions, sine and cosine, can be de?ned in

terms of the unit circle--the set of points in the Euclidean plane of distance one from

the origin. A point on this circle has coordinates +frv w> vlq w,, where w is a measure (in

radians) of the angle at the origin between the positive {-axis and the ray from the ori-

gin through the point measured in the counterclockwise direction. The other four basic

trigonometric functions can be de?ned in terms of these two--namely,

wdq {

@

vlq { frv {

vhf

{

@

4 frv

{

frw {

@

frv { vlq {

fvf

{

@

4 vlq {

For

3

?

w

?

5,

these

functions

can

be

found

as

a

ratio

of

certain

sides

of

a

right

triangle

that has one angle of radian measure w.

Trigonometric Functions

The symbols used for the six basic trigonometric functions--vlq, frv, wdq, frw, vhf, fvf--are abbreviations for the words cosine, sine, tangent, cotangent, secant, and cosecant, respectively.You can enter these trigonometric functions and many other functions either from the keyboard in mathematics mode or from the dialog box that drops down

when you click

or choose Insert + Math Name. When you enter one of these

functions from the keyboard in mathematics mode, the function name automatrically

turns gray when you type the ?nal letter of the name.

90 Chapter 4 Trigonometry and Complex Numbers

Note Ordinary functions require parentheses around the function argument, while trigonometric functions commonly do not. The default behavior of your system allows trigonometric functions without parentheses. If you want parentheses to be required for all functions, you can change this behavior in the Maple Settings dialog. Click the Definition Options tab and under Function Argument Selection Method, check Convert Trigtype to Ordinary. For further information see page 126.

To ?nd values of the trigonometric functions, use Evaluate or Evaluate Numerically.

L Evaluate

vlq

6

7

@

s

4 5

5

vlq +4, @ vlq 4

vlq 93

@

s

4 5

6

L Evaluate Numerically

vlq

6

7

@

= :3:44

vlq +4, @ = ;747:

vlq 93 @ = ;9936

The notation degrees: vlq 63

you @

u= @5,

, before applying Solve +

Numeric

or

you

may

get

a

solution

greater

than

5

.

Specifying

these

intervals

gives

the

solution

e @ 45= @ = @ =8 ................
................

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