Trigonometry Book Part 2

Trigonometry

TRIGONOMETRY ........................................................................................................................................................1

AUTHORS...................................................................................................................................................................3

PREREQUISITES AND BASICS .....................................................................................................................................4

INTRODUCTION..........................................................................................................................................................4

IN SIMPLE TERMS .......................................................................................................................................................4

Simple introduction..............................................................................................................................................4

Angle-values simplified ........................................................................................................................................5

RADIAN AND DEGREE MEASURE ...............................................................................................................................7

A Definition and Terminology of Angles..............................................................................................................7

The radian measure .............................................................................................................................................7

Converting from Radians to Degrees...................................................................................................................9

Exercises ..............................................................................................................................................................9

THE UNIT CIRCLE....................................................................................................................................................11

TRIGONOMETRIC ANGULAR FUNCTIONS..................................................................................................................12

Geometrically defining sin and cosine ...............................................................................................................12

Geometrically defining tangent ..........................................................................................................................13

Domain and range of circular functions ............................................................................................................14

Applying the trigonometric functions to a right-angled triangle .......................................................................14

RIGHT ANGLE TRIGONOMETRY ...............................................................................................................................14

GRAPHS OF SINE AND COSINE FUNCTIONS ..............................................................................................................16

GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS....................................................................................................17

INVERSE TRIGONOMETRIC FUNCTIONS....................................................................................................................21

The Inverse Functions, Restrictions, and Notation ............................................................................................21

The Inverse Relations.........................................................................................................................................21

APPLICATIONS AND MODELS...................................................................................................................................23

Simple harmonic motion ....................................................................................................................................23

ANALYTIC TRIGONOMETRY.....................................................................................................................................26

USING FUNDAMENTAL IDENTITIES ..........................................................................................................................26

SOLVING TRIGONOMETRIC EQUATIONS...................................................................................................................28

Basic trigonometric equations ...........................................................................................................................28

Further examples ...............................................................................................................................................33

SUM AND DIFFERENCE FORMULAS ..........................................................................................................................35

Cosine Formulas ................................................................................................................................................35

Sine Formulas ....................................................................................................................................................35

Tangent Formulas ..............................................................................................................................................35

Derivations.........................................................................................................................................................35

MULTIPLE-ANGLE AND PRODUCT-TO-SUM FORMULAS ...........................................................................................37

Multiple-Angle Formulas ...................................................................................................................................37

Proofs for Double Angle Formulas....................................................................................................................37

ADDITIONAL TOPICS IN TRIGONOMETRY.................................................................................................................38

LAW OF SINES..........................................................................................................................................................38

LAW OF COSINES .....................................................................................................................................................39

VECTORS AND DOT PRODUCTS................................................................................................................................41

TRIGONOMETRIC FORM OF THE COMPLEX NUMBER................................................................................................41

TRIGONOMETRY REFERENCES .................................................................................................................................42

TRIGONOMETRIC FORMULA REFERENCE .................................................................................................................42

TRIGONOMETRIC IDENTITIES REFERENCE................................................................................................................42

Pythagoras .........................................................................................................................................................42

Sum/Difference of angles ...................................................................................................................................42

Product to Sum...................................................................................................................................................43

Sum and difference to product ...........................................................................................................................43

Double angle......................................................................................................................................................433

Half angle...........................................................................................................................................................44

Power Reduction................................................................................................................................................44

Even/Odd............................................................................................................................................................44

Calculus .............................................................................................................................................................45

NATURAL TRIGONOMETRIC FUNCTIONS OF PRIMARY ANGLES ...............................................................................46

LICENSE...................................................................................................................................................................48

GNU Free Documentation License ....................................................................................................................48

0. PREAMBLE ...................................................................................................................................................48

1. APPLICABILITY AND DEFINITIONS ..........................................................................................................48

2. VERBATIM COPYING...................................................................................................................................49

3. COPYING IN QUANTITY..............................................................................................................................49

4. MODIFICATIONS .........................................................................................................................................49

5. COMBINING DOCUMENTS.........................................................................................................................50

6. COLLECTIONS OF DOCUMENTS ..............................................................................................................50

7. AGGREGATION WITH INDEPENDENT WORKS .......................................................................................51

8. TRANSLATION..............................................................................................................................................51

9. TERMINATION..............................................................................................................................................51

10. FUTURE REVISIONS OF THIS LICENSE..................................................................................................51

External links .....................................................................................................................................................51

Authors

Lmov, Evil Saltine, JEdwards, llg, Programmermatt, Alsocal (Wikibooks Users)4

Prerequisites And Basics

To be able to study Trigonometry sucessfully, it is recommended that students complete;

Geometry, Algebra I and Algebra II prior to digging in to the course material.

It is helpful to have a graphing calculator and graph paper on hand to be able to follow along as

well. If one is not available software available on sites such as http://www.graphcalc.com/ may

be helpful.

Introduction

Trigonometry is an important, fundamental step in math education. From the seemingly simple

shape, the right triangle, we gain tools and insight that help us in further practical as well as

theoretical endeavors. The subtle mathematical relationships between the right triangle, the

circle, the sine wave, and the exponential curve can only be fully understood with a firm basis in

trigonometry.

In simple terms

This page is intended as a simplified introduction to trigonometry. (This article is not always

correctly formulated in mathematical language.)

Simple introduction

If you are unfamiliar with angles, where they come from, and why they are actually required, this

section will help you develop your understanding.

In principle, all angles and trigonometric functions are defined on the unit circle. The term unit in

mathematics applies to a single measure of any length. We will later apply the principles gleaned

from unit measures to a larger (or smaller) scaled problems. All the functions we need can be

derived from a triangle inscribed in the unit circle: it happens to be a right-angled triangle.

A Right Triangle5

The center point of the unit circle will be set on a Cartesian plane, with the circle's centre at the

origin of the plane — the point (0,0). Thus our circle will be divided into four sections, or

quadrants.

Quadrants are always counted counter-clockwise, as is the default rotation of angular velocity ω

(omega). Now we inscribe a triangle in the first quadrant (that is, where the x- and y-axes are

assigned positive values) and let one leg of the angle be on the x-axis and the other be parallel to

the y-axis. (Just look at the illustration for clarification). Now we let the hypotenuse (which is

always 1, the radius of our unit circle) rotate counter-clockwise. You will notice that a new

triangle is formed as we move into a new quadrant, not only because the sum of a triangle's

angles cannot be beyond 180°, but also because there is no way on a two-dimensional plane to

imagine otherwise.

Angle-values simplified

Imagine the angle to be nothing more than exactly the size of the triangle leg that resides on the

x-axis (the cosine). So for any given triangle inscribed in the unit circle we would have an angle

whose value is the distance of the triangle-leg on the x-axis. Although this would be possible in

principle, it is much nicer to have a independent variable, let's call it phi, which does not change

sign during the change from one quadrant into another and is easier to handle (that means it is

not necessarily always a decimal number).

!!Notice that all sizes and therefore angles in the triangle are mutually directly proportional. So

for instance if the x-leg of the triangle is short the y-leg gets long.

That is all nice and well, but how do we get the actual length then of the various legs of the

triangle? By using translation tables, represented by a function (therefore arbitrary interpolation

is possible) that can be composed by algorithms such as taylor. Those translation-table-functions

(sometimes referred to as LUT, Look up tables) are well known to everyone and are known as

sine, cosine and so on. (Whereas of course all the abovementioned latter ones can easily be

calculated by using the sine and cosine).

In fact in history when there weren't such nifty calculators available, printed sine and cosine

tables had to be used, and for those who needed interpolated data of arbitrary accuracy - taylor

was the choice of word.

So how can I apply my knowledge now to a circle of any scale. Just multiply the scaling

coefficient with the result of the trigonometric function (which is referring to the unit circle).

And this is also why cos(φ)

2

+sin(φ)

2

 = 1, which is really nothing more than a veiled version of

the pythagorean theorem: cos(φ)=a;sin(φ)=b;a

2

 + b

2

 = c

2

, whereas the c = 1

2

 = 1, a peculiarity

of most unit constructs. Now you also see why it is so comfortable to use all those mathematical

unit-circles.

Another way to interprete a angle-value would be: A angle is nothing more than a translated

'directed'-length into which the informati