Table of Contents

Chapter 1

For information:

Fred W. Duckworth, Jr.

c/o Jewels Educational Services

1560 East Vernon Avenue

Los Angeles, CA 90011-3839

E-mail: admin@trinitytutors.com

Website: www.trinitytutors.com

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INSTRUCTOR: Mr. Will

INTRODUCTION TO TRIGONOMETRY

I avoided trigonometry in high school, having heard from peers that it was extremely difficult (I took geometry instead).  When I could avoid it no longer (college) I discovered that it wasn't so difficult after all. Nonetheless, I dropped the class half way through due to unspeakable boredom.  However, the point I want to make is that I now realize even though I was receiving all A's and B's on my assignments, midway through I still had no genuine understanding of what in the world trigonometry was all about.

The way it was taught, trigonometry simply amounted to a bunch of abstract functions and definitions into which I learned to plug and chug values – thoughtlessly mimicking various solving methods to successfully complete problems involving concepts about which I had no clue.

Consequently, when I set about writing this textbook, I had to go back and learn trigonometry "for real."

In the hope that you'll established an authentic understanding of trigonometry from the very beginning, I have purposely chosen not to copy the format established by the other textbooks.  I am afraid that doing so would mean you too might experience the same fog that constituted my introduction to the topic.

So, rather than rely on past textbooks (characterized by poorly emphasized concepts -- often leading to problems  understanding foundational principles), I have chosen instead to select alternate sources of material and vary this book's organization in order to improve instruction.

U

T U T O R S

TRINITY

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Trigonometry

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Trigonometry

4Trigonometry

5Trigonometry

Lesson 1

It amazes me that or I got halfway through my own class without it having ever dawned on me that trigonometry is all about the relationships that exists between the sides of a (right) triangle with respect to a given angle.

I therefore contend that, to understand trigonometry, you have to first understand those relationships.

A unit-cirlce (a circle that has a radius of one unit and has its center at the origin of a rectanglar coordinate system) is a handy little tool for helping us to accomplish this goal.

We begin by constructing our unit-circle on an x-y coordinate plane. Let’s give it a central angle θ (pronounced theta) and we’ll let sine be the top half of a chord that is bisected by the x-axis (a chord is a line segment that joins two points on a curve).

Now, let’s draw a triangle in our unit circle as shown above -- one with a hypotenuse equal to the radius. Note that the side adjacent to    θ lies along the x-axis, and the (x, y) coordinates at the outer end of the hypotenuse identify the lengths of the sides adjacent to and opposite of    θ respectively. Such a triangle is called a unit-triangle.

Our half-chord has been colored red for easy identification. Its name – sine – is derived from the Latin word sinus, which took the place of the Arabic word that took the place of the Hindu word meaning “half-chord.”

Since trigonometry is all about the relationships between the sides of a (right) triangle, what we now need is another side of our unit-triangle that will relate to sine in such a way as to define it. Or to put it another way, we need some side of our unit-triangle whose relationship with sine will return sine right back to us.

In other words, we need is ratio (the relation between two quantities expressed as the quotient of one divided by the other) of sine:1 – one in which we can divide sine by 1.

^

>

<

V

r = 1

(x, y)

←  sine

θ

The term "sine" is derived from the Latin word, sinus, which took the place of the Arabic word that took the place of a Hindu word meaning "half-chord."

Now, we need some other side of our triangle that relates to sine in such a way as to allow us to "define" sine. The side which does that for us is the hypotenuse, (since it forms a ratio of sine:1).

Although we initially labeled sine as the half-chord directly across from angle theta, in actuality, sine is more accurately described as the ratio of the half-chord to the hypotenuse.

CLASS: The Magnificent Unit Circle!

side opposite

hypotense

sine θ =

INSTRUCTOR: Mr. Will

6Trigonometry

Lesson 2

So, in reality, sine is simply the name assigned to a certain function (a rule of correspondence between two sets such that there is a unique element in the second set assigned to each element in the first).

We wouldn’t want to have to go around saying “side opposite the angle of consideration divided by the hypotenuse” all the time, so we simply say . . .“sine” instead.

Sine is abbreviated this way: sin

The element in the function’s first set is the measure of the angle being considered. The rule that is applied to the first element (the angle measure) is “divide the length of side opposite the angle by the length of the hypotenuse.”

Since sin is a function, rather than write f(x), we write sin(x) instead.

And we drop the parentheses so that the standard notation usually looks like this:

sin x.

Nevertheless, always remember that sin is a function, with the “rule” being: side opposite divided by the hypotenues.

EXAMPLE:

Given the lengths of the sides of the triangle below, here is how we determine the sine of 32.4°.

Since sin is obtained by dividing the length of the side that is opposite the angle of

consideration by the hypotenuse, we have   sin (32.4°) =              = 0.6

That was pretty simple don’t you think? (Note that sin is a pure number.)

The term "sine" is derived from the Latin word, sinus, which took the place of the Arabic word that took the place of a Hindu word meaning "half-chord."

Now, we need some other side of our triangle that relates to sine in such a way as to allow us to "define" sine. The side which does that for us is the hypotenuse, (since it forms a ratio of sine:1).

Although we initially labeled sine as the half-chord directly across from angle theta, in actuality, sine is more accurately described as the ratio of the half-chord to the hypotenuse.

CLASS: Solving the Mystery of Sine

side opposite

hypotense

sine θ =

5'

4'

3'

32.4º