Trigonometry 101

TrinityTutors.com

Trigonometry 101

TrinityTutors.com

Trigonometry 101

Jewels Educational Services

Sine

T R I G O N O M E T R Y

Lesson 2

We are now going to establish the ratio that exists between a given angle and a particular side of a right triangle.

To do so, we draw our right triangle inside of a unit circle as illustrated below. The angle of consideration (angle theta) goes in the middle.

The side adjacent to angle theta lies along the x-axis.

The side opposite angle theta is formed by a half-chord.

The triangle's hypotenuse (the side opposite the right angle) is equavalent to the radius. This is an extremely important fact for reasons that will soon be appearant. We call such a triangle, a unit triangle.

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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Table of Contents

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Let's begin by establishing a basic understanding of just what trigonometry is. Stated as simply as possible, trigonometry is a specialized area of geometry that involves the properties of, and relations between, the parts of a triangle. Early on, it was concerned exclusively with ratios existing between the angles and sides of right triangles. The term is derived from the Greek word trigonon, meaning triangle, and from the Greek word metron, meaning measurement.

So, for our purposes, let's define trigonometry as a study of the ratios that exist between the sides and the angles of a triangle.

This may not be an official definition, but it will enable us to conceptualize what in the world we are dealing with here, which is my main concern at this point.

Nowadays, trigonometry is often a source of significant grief, which is a real shame given that it is an indispensable tool among architects, surveyors, and astronomers -- especially spherical trigonometry, which is concerned specifically with the study of triangles on the surface of a sphere, such as the Earth.

To minimize such heartache, whenever possible we will examine trigonometric concepts geometrically (using drawings) rather than simply in terms of abstract algebraic equations.

Knowledge of what are referred to as sine, cosine, and tangent, along with the ability to prove certain basic identities related to them, are especially important if one intends to study calculus, advanced mathematics, engineering, physics, or other sciences, so there is indeed great value in knowing trigonometry.

It is my sincere hope that once you knows how all of the pieces hang together, you will be able to combine trigonometric identities in different ways to derive most any formula needed to solve practical, real-world trigonmetry problems. To begin this process, we will look at something called the "unit circle" which will enable us to learn the six foundational trigonometric functions without relying on rote memorization.

The Magnificent Unit Circle!

T R I G O N O M E T R Y

Lesson 1

In its infancy, trigonometry was treated mainly as an adjunct to astronomy. Only later were trigonometric concepts defined strictly in terms of right triangles without the use of circles. So, to make the most sense out of trigonometry, it seems only natural for us to take that same approach. To do so, we will use the fabulous “unit circle” to enable us to conceptualize the six basic trigonometric functions.

The unit circle provides us with insight into the origin of trigonometric functions -- or ratios -- and how they their uses in the past relate to how they are currently used and taught today. It allows us to see and make connections between trigonometry and real-life, and to understand from where the basic trigonometric concepts came.

We define a unit as a basic amount that is used to count or measure. In this case, we are using it to measure.

A unit-cirlce is so named because it has a radius that measures one unit in length.

It is drawn with its center at the origin of an x-y coordinate plane.

As is the case with everything else found at this website, this book came about as the result of my having discovered an inadequate number of Internet sites able to meet my need for free, high-quality curriculum materials. Required to ultimately design most of the materials I needed for myself, I decided to make them available to everyone.

The specific goal of this particular book is to enable beginning trigonometry students to thoroughly understand tigonometric functions, formulas, and identities and how they work.

Unfortunately, the way I was taught trigonometry amounted to a bunch of abstract formulas into which I learned to plug and chug values -- thoughtlessly mimicking the examples in the textbook to successfully solve problems involving concepts about which I hadn’t a clue. I want better than that for you!

In the hope that you will establish an authentic understanding of trigonometry, I have purposely chosen not to follow a traditional format, since doing so would merely increase the likelihood of your experiencing the same fog that constituted my early education in the subject. Moreover, in accordance with my instructional philosophy, every attempt has been made to avoid teaching trigonometry abstractly, for I agree with Stan Brown of Trig without Tears fame that attempting to memorize all of the trigonometric identities by rote leads to disasterous results.

However, while Mr. Brown feels some rote memorization is necessary, I have found that even this minimum amont of “cryptography” may impeed progress toward a firm grasp of the basic trigonometric functions.

Consequently, this textbook will present basic information using a concrete approach that mimics the way the discipline originally developed.

A Closer Look at “Sine”

Though sine is actually a ratio, for now we will conceptualize it as a line segment whose length is dependent on the size of our unit circle's central angle. This line segment is the half-chord directly across from the central angle.

cos θ =

length of the side adjacent θ

hypotenuse

We continue our study of trigonometry by examining the relationships existing between the sides of a unit triangle with respect to central angle θ (theta). Therefore, we will give the sides of our unit triangle names that represent their relationship to that angle.

The first side we want to look at is the side that is opposite angle theta. We have colored it red for easy identification.

Noticed that this side is formed by a half-chord. (A chord is a line segment that joins two points on a curve.) Its name is derived from a Latin word, which took the place of an Arabic word, which took the place of a Hindu word that meant “half-chord.”

So, though the Latin word sinus does not mean half-chord per se, (the original Hindu word did mean half-chord), we will call our first ratio sine, because it involves a half-chord -- or sinus -- from which the term sine is derived

Let's take a look at that one more time.

We noticed that our “sinus” (or our half-chord) is opposite our angle of consideration-- angle θ. We also noted that if we wanted a ratio that would produced a number identical to the length of “sinus,” we would have to divide “sinus” by 1.

Hmmm, let’s see here now . . .

We considered whether there was some other part of our unit triangle that would allow us to do that, and as a matter of fact, there was. The hypotenuse would, because its length is exactly 1 unit.

So, dividing “sinus” (we’ll start calling it sine now) by the hypotenuse would give us sine right back, and since sine is the side of the triangle that is opposite our angle of consideration, we could say that the sine of angle θ is equal to the side that is opposite that angle divided by the hypotenuse . . .

So, given any right triangle, we can find the sine for either of its acute angles by dividing the side opposite that angle by the hypotenuse.

NOTE; While we have given the opposite side of an angle the name “sine” it is critical that you relize that the sine of an angle does not actually refer directly to the side opposite the angle, but to the ratio between the side opposite that angle and the corresponding hypotenuse!

r = 1

length of the side opposite

hypotenuse

< sine

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

angle by the hypotenuse, we have sin(A) = = 0.6402

That was pretty simple, wasn’t it?

6 in.

7.81 in.

5 in.

2 ILLUSTRATION Applying the Concept

Sine is simply a name assigned to a certain relationship. Instead of going around saying “side opposite divided by hypotenuse” all the time, we simply say: sine. In other words, sine is a 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.

It abbreviated this way: sin

The element in the first set is the measure of some angle within a triangle. The rule that is applied to the first element (the angle measure) is “divide the triangle’s hypotenuse by the side opposite the angle of consideration.”

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 . . . divide the opposite side by the hypotenues.

EXAMPLE:

If the lengths of the sides of the triangle below were as follows, here is how we would determind the sine of Angle A.

cosine

32°

90°

58°

1.0

0.8

0.6

r =

3 EXPLANATION The Meaning of Cosine

Figure 1.4

Figure 2.3

Figure 1.3

3 EXPLANATION The Meaning of Tangent

To calculate the tangent of an angle, divide the length of the side that is opposite to that angle by the side that is adjacent to it.

Figure 1.5

Tangent (a line, curve, or surface making contact with another line, curve, or surface at a single point; touching but not intersecting) is from the Latin tangere, meaning: to touch.

What about tan θ? Well, since the green line segment is tangent to the unit circle, you might guess that its length is tan θ, and you’d be right. Note that the triangles formed by the horizonatl radius along with the gray and green line segments is similar to the enclosed unit triangle.

Therefore, dividing the green line segment by the horizontal radius will result in the same ratio as dividing the red line segment by the orage line segment.

Dividing the tangent by the radius gives us the ratio that is equivalent to the tangent (since the radius = 1).

Moving to our unit triangle, that same ratio is given by the side opposite divided by the side adjacent, or sin divided by cos,

Hence, we have just determined that... tan θ = sin θ / cos θ

Since the green and black vetical line segments below are both parallel and intersected by the same line segment (the hypotenuse), their corresponding angles — both marked theta in the diagram — are congruent. Consequently, the white triangle on the top has to be similar to the yellow triangle on the bottom.

That means that the violet line segment (the one that is tangent to the top of the unit circle) divided by the vertical radius (along the y-axis) will yield the same ratio as the horizontal radius (along the x-axis) divided by the green line segment.

In other words, the "cotangent" divided by 1 = 1 divided by the tangent, or cot = 1/tangent. (It's almost like outlining a square.)

That makes sense: the cotangent is tangent to the unit circle, and is the tangent of the complement of angle θ, and is therefore the cotangent of the original angle θ.

Figure 3.1

< tangent

side opposite θ

side adjacent θ

tan θ =

tan θ

cot θ =

cos θ

sec θ =

(x,y)

Figure 1.5

Figure 1.6

Finally, looking at our unit trianlge and the top (transparent) similar triangle one last time, we can also see that . . .

the cosecant divided by the vertical radius (along the y-axis) = hypotenuse (of the unit triangle) divided by sine

cos θ / 1 = 1 / sin θ

cos θ = 1 / sin θ

(Since I can't draw red arrows on this one I think of it this way: The “first seeker” takes the escalator down in the elevator up on his search. His “co-seeker” takes the escalator up in the elevator down on his search.)

< secant

< cosecant

cotangent

Special Triangles

sin θ

csc θ =

sine θ =

r = 1

We begin using an x-y coordinate plane on which we construct a unit-circle, defined as a circle

with a radius of 1 unit and center at the origin of a rectangular coordinate system.

The unit circle is taught using important degree

measures (30, 60, 45, 90 degrees, etc) and then students are asked to put them into radian

form as well.

This leaves them with the given diagram as a tool.

The textbooks then incorporate this tool, the labeled unit circle, into the next lesson to help students learn sine, cosine and tangent.

Sine, cosine, and tangent are first introduced as functions, expressed as sin(t), cos(t), and tan(t).

Their initial definitions are as follows:

sin(t) = y cos(t) = x tan(t) = y/x

Therefore, their function values are expressed in

terms of coordinates of a point on the unit circle, sine being the y-coordinate, cosine the x-coordinate, and

Let tangent be a line parallel to sine and tangent to the unit circle at the x-axis, and extend hypotenuse until it meets tangent.

We have the tangent function related to the tangent line in the same way as the sine function is to the half-chord.

Introduction

T R I G O N O M E T R Y

A unit circle is a handy little tool for helping us to accomplish our first goal, which is to understand the relationships that exist between the sides of a right triangle.

But, in place of the word relationship, we're going to begin using the word ratio instead. And the name of the first ratio that we are going to examine is called sine.

Concept Review:

Directions: If you cannot answer any of the following questions, you need to go back and study the material until you internalized it.

1. What is trigonometry?

Figure 1.1

Figure 1.2

Figure 2.1

Figure 2.2

← sine

In a unit-circle with central angle theta and radius = 1, let sine be the half-chord, bisected by the x-axis.

You probably know, a ratio is a division problem. So, in oder for our ratio and are half cord to be equivalent, we must divide the half cord by one. And what was equal to one? Oh, yes, it was our hypotenuse. So if we divide sine by hypotenuse, we still end up with sine. So the Sign of theta is defined as the side opposite divided by the hypotenuse.

length of the side opposite

hypotenuse

sine θ =

Sure, simple, but who cares? Well, since we know that an angle of % had a sine of 0.6402, if I have a the Pico Army built a wall 100 feet tall and our army wants to scale it on laders tilted at 45%, how long must our ladders be.

Using algebra 100 divided by x = 0.6402

x = divided by 100 = 1 divided by 0.6402

x = 0.6402 x 100

Our laders have to be 60feet long.

cliff to cliff.

5 in.

7.81 in

Figure 2.1

Cosine

We've got a problem however, because we want to remain consistent and describe all of our ratios in terms of angle theta.

Well, wait a minute!

The side that is opposite the other angle, our orange side, is the same thing as the side that is adjacent to angle theta.

So, if we divide the side that is adjacent to angle theta by the hypotenuse, we get the sine of the other angle.

Therefore, let's call the sine of the other angle the cosine of angle theta, and let's define the cosine of angle theta as side adjacent divided by the hypotenuse. In the example below, the side next to the angle of consideration is 0.8 units long, and . . .

0.8 divided by 1.0 = 0.8, so . . . cos θ = 0.8

Cosine

T R I G O N O M E T R Y

Figure 2.1

Lesson 3

What if we want to change our angle of consideration? What if you want to find the sine for the other angle?

Since we gave the name sine to the side opposite and angle of consideration, we find that the sine of the other angle lies along the y-axis. (We have colored it orange for easy identification.)

We know that a right angle has a measure of 90°, and that all the angles of a triangle add up to 180°. So it follows that the measure of our last angle will be 90 - θ.

This means that the sine of 90 - θ = our "orange side" divided by the hypotenuse.

So, the cosine function first arose from the need to compute the sine of the angle of consideration's complementary angle. (Complementary angles are angles such that the sum of their two measures is 90°).

We found that we can get the sine of theta's complementary angle by dividing its own side opposite (the orange side) by the hypotenuse.

However, that's the exact same thing as dividing angle theta's side adjacentby the hypotenuse.

Therefore, for the sake of consistency, instead of calling this ratio the sine of the complementary angle, we call it the cosine of theta, and we define it as side adjacent divided by the hypotenuse.

Nonetheless, keep in mind that the cosine of theta, and the sine of theta's complementary angle are the exact same ratio!

The consine is abbreviated cos.

To calculate the cosine of an angle, divide the length of the side that is adjacent to that angle (next to it) by the hypotenuse.

Concept Review:

Directions: If you cannot answer any of the following questions, you need to go back and study the material until you internalized it.

What is trigonometry?

What is the ratio that defines sine?

What is the ratio the defines cosine?

In a right traingle, what is the relationship between the sine of a given acute angle, and that angle's complement?

3 EXPLANATION The Meaning of Sine

Tangent

T R I G O N O M E T R Y

Cotangent

T R I G O N O M E T R Y

Secant

T R I G O N O M E T R Y

Cosecant

T R I G O N O M E T R Y

Can you see what else this illustrates?

If the cotangent divided by one is equivalent to one divided by the tangent, what else can we say?

Well, for one thing, that means that the cotangent divided by one is equivalent to side adjacent divided by side opposite.

And one more thing . . .

It also means that cotangent divided by one is also equivalent to cosine divided by sine.

Can you see all that?

Figure 3.1

tan θ

cot θ =

cotangent

Secant is from the Latin root Secare, meaning to cut. It is a proper name for a segment that cuts through the circle.

Looking at the two similar triangles below (the unit triangle and the larger triangle formed by the secant, the tangent, and the horizontal radius) we note that the ratio of the secant divided by the horizontal radius (on the larger triangle) yields the same ratio as does the unit triangle's hypotenuse divided by the cosine. (I see a large fish with its pink mouth open wide “seeking” some food to eat.)

Since the horizontal radius = 1, the ratio that gives us the secant is: secant divided by a horizontal radius.

The equivalent ratio on our unit triangle is: hypotenuse divided by side adjacent, and since the hypotenuse is equal to one, we have:

secant = 1/side adjacent, and since side adjacent is cosine, we have:

sec θ = 1/cos