Gauss-Jordan Elimination

Lesson 1

The Mystersy of “Sine”

To understand trigonometry, we must first understand the relationships that exist between the sides of a right triangle. 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.”

Now, all we need is another side of our unit-triangle to which we can relate sine and still end up with sine.

Hmm . . . Well now, let’s see here . . . what we need is a ratio (the relation between two quantities expressed as the quotient of one divided by the other) of sine:1.

So, we need some other side of our triangle that will allow us to do divide sine by 1. But, does such a side exist?

Yes! As a matter of fact, it does. The hypotenuse will work, since the hypotenuse is equivalent to the radius, and radius = 1.

So, dividing sine by the hypotenuse will give us sine right back.

Remember that trigonometry is all about the relationships that exist between the sides of a right triangle with respect to a given angle. So, although we initially labeled sine as the half-chord directly across from θ, in actuality, sine is more acurately described as the ratio between the half-chord and the hypotenuse.

And since the half-chord (the line segment we've been calling sine) is opposite θ, in the final anlysis, what we actually have is . . .

Trigonometry

r = 1

CLASS: The Magnificent Unit Circle!

INSTRUCTOR: Mr. Will

The term "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."

r = 1

(x, y)

Looking at “Cosine”

Our next function is the cosine function, which first arose from the need to compute sin for the complementary angle (an angle related to another angle so that the sum of their two measures is 90°).

Note that we can calculate sin for the comlement of 32.4° (sin for 57.6°) by dividing the side opposite the 57.6° angle by the hypotenuse. But, the side opposite the 57.6° angle is the same side as the one that is adjacent to our original angle.

Well then, lets not go to the complementary angle after all. Let’s stick with our original angle and give the side adjacent to it a name of its own. We’ll call it cosine and we’ll color it orange for easy identification. Let’s abbreviate it this way: cos.

Now we need another side of our unit-triangle to which we can relate cosine and still end up with cosine, so once again, lets divide by our hypotenuse.half-chord.”

← sine

(x, y)

sine θ =

length of the side opposite θ

hypotenuse

We have just unlocked the mystery of sine!

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.)

EXAMPLE:

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

3 ft.

5 ft.

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.

Figure 1.2

32.4°

Applying the Concept!

Going back to Figure 1.2 we determine cos (32.4°) by dividing the side adjacent to that angle by the hypotenuse, which amounts to dividing 4 by 5.

Hence . . . cos (32.4°) = = 0.8

A Look at “Tangent”

Our next function is the tangent (a line, curve, or surface making contact with another line, curve, or surface at a single point; touching but not intersecting) function, derived from the Latin tangere, meaning “to touch.”

Going back to our unit-circle, let tangent be a line segment parallel to sine and tangent to the unit-circle at the x-axis, and let the hypotenuse be extended until it meets tangent. (Tangent has been colored green for easy identification.)

Now we need some other side of the triangle formed by the x-axis, the extended hypotenuse, and tangent to which we can relate tangent and still end up with tangent. The only way is to divide tangent by the radius (along the x-axis).

Note that the triangle formed by the x-axis, the hypotenuse, and tangent is similar (having corresponding angles equal and corresponding line segments proportional) to the triangle formed by the x-axis, the hypotenuse, and sine.

Therefore, dividing tangent by the radius will result in the same ratio as dividing sine by cosine. Let’s abbreviate tangent this way: tan.

Hence we have . . .

A Look at “Cotangent”

Our next function is the cotangent function, named after the line segment tangent to the top of our unit circle, hence the name: cotangent. (Cotangent has been colored violet for easy identification.)

Since sine and the y-axis are parallel to one another and intersected by the same line segment, their corresponding angles -- both lebeled θ -- must be congruent. Thus, the top triangle (the one that includes cotangent) must be similar to the bottom triangle (the one that includes tangent).

That means that cotangent divided by the radius (along the y-axis) will yield the same ratio as the radius (along the x-axis) divided by tangent.

In other words, cotangent divided by 1 = 1 divided tangent . . .

So, we have . . . cot θ =

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.

A Look at “Secant”

Our next function is the secant function. Secant is from the Latin root secare, meaning “to cut.” The name is fitting for a segment that cuts through the circle. We will color secant maroon for easy identification.

Looking at the two similar triangles above (the one that includes tangent and the one that includes sine) we note that comparing the hypotenuse of the larger triangle to the hypotenuse of the smaller one will yield the same ratio as comparing the radius along the x-axis to cosine.

Consequently, we have secant divided by 1 = 1 divided by cosine

In other words . . .

A Look at “Cosecant”

And finally, our last function is the cosecant function.

Thise time our top triangle (the one that includes cotangent) is compared to the smaller unit-triangle (the one that includes sine).

We will color cosecant (so named because it is the secant of the complement of angle θ, and is therefore the cosecant of the original angle θ) brown for easy identification.

Note that the ratio between cosecant and the radius (along the y-axis) is the same as the ratio between the radius (along the hypotenuse) and sine.

Hence, we have cosecant divided by 1 = 1 divided by sine.

Therefore . . .

Special Angles

You need to know the function values of certain special angles and you really need to memorize them because you’ll use them so often that deriving them or looking them up every time would really slow you down.

Let's begin with Functions of 45°.

Look at the 45°-45°-90° unit-triangle illustrated below.