Project in Math

Chapter 6
Circular Functions and Trigonometry

Submitted to: Mrs. Velasco
Submitted by: Angelo Jet Manarin
IV – Bravo

Angles & Their Measures
A definition of an angle would be that an angle is the union of two rays that have the same endpoint. The sides of the angles are the two rays, while the vertex is their common endpoint. stands for an angle. You can put it in front of three letters which represent points. The first and third letters represent points on each of the rays that form one of the sides. The middle letter represents the vertex. As you can see in the diagram, each point is represented in the written form. The letters can go either way - that is, first and last letter are interchangeable. So, that angle could either be ABC or CBA. Since B is the vertex, it is always in the middle of the two letters. You can also name an angle by just the letter of its vertex. So, for the example in the picture, the angle could also be labeled B. That's only if there are no other angles that share the same vertex. There is a third way to label angles. In the third way, each angle is designated with a number, so the example could be labeled 1 or 2 or whatever you wanted.
Angles are measured in degrees. The number of degrees tell you how wide open the angle is. You can measure angles with a protracter, and you can buy them at just about any store that carries school items. Degrees are marked by a ° symbol. For those of you whose browsers can't interpret that, a degree symbol looks like this: . I tend to just write it out instead of using the symbol because it's quicker on the computer. There are up to 360 degrees in an angle. As you can see in the picture below, the 360 degrees form a circle.
There are a few more basic things you should know about angles. First of all, the space inside an angle of less than 180 degrees, is a convex set, while the space outside of one is a nonconvex set. The opposite is true for an angle of more than 180 degrees (but less than 360 degrees). The side of an angle that is started at would be called the initial side, and the side that an angle ended at would be called the terminal side. The measure of ABC is written mABC.
When measuring angles, you usually go counterclockwise, starting where the 3 would be on a clock. That would be called a zero angle because there is nothing in it - just a single ray going directly to the right. The next important type of angle is called the acute angle. An acute angle is an angle whose measure is inbetween 0 and 90 degrees. An example would be the 45 degree angle in the picture. The next important type of angle is the right angle. This is probably the most important type of angle there is because of all the spifty things that you can do with one. I won't go into all of them here. (I have to save something for later articles!) A right angle is an angle whose measure is exactly 90 degrees. Continuing around the circle, next is theobtuse angle. An obtuse angle is an angle whose measure is inbetween 90 and 180 degrees. The 135 degree angle in the diagram is an example. The last major kind of angle is the straight angle. A straight angle is an angle that measures exactly 180 degrees. Thus the name - the two rays form a straight line. A negative angle is also possible. This just means that you go clockwise instead of counterclockwise.
A lot of geometry teachers don't go beyond that, at least at first. There isn't much else left to explain, but I'll give it a shot. After straight angles, there aren't any more special angles that you need to know about. A 360 degree angle is an angle that does a full circle. It looks just like a zero angle, but instead of having no degrees, it has 360 of them. (Duh. You can't get more basic than that!)
It is possible to have an angle with more than 360 degrees. To find out what it looks like, all you do is subtract 360 from it until you have an angle less than or equal to 360. (What?! You want an example? C'mon, you people...) For example, if you have an angle that is 546 degrees, you subtract 360 from 546 to get 186. Thus, the angle is the equivalent of a 186 degree angle.
There are a few more terms that you should also know. Supplementary angles are two angles whose measures combined equal 180 degrees. Complementary angles are two angles whose measures combined equal 90 degrees. Two non-straight and non-zero angles are adjacent if and only if a common side is in the interior of the angle formed by the non-common sides. A linear pair is a pair of adjacent angles whose non-common sides are opposite rays. Vertical angles are two angles that have a common vertex and whose sides form two lines. is a bisector of DAC if and only if is in the interior of DAC and mDAB = mCAB.
Angle Measure Postulate
A) Unique Measure Assumption: Every angle has a unique measure. The measure could be infinite or negative.
B) Two Sides of a Line Assumption: Given any ray and any number x, there are unique rays and such that intersects ray and mBEA = mCEA = x.
C) Zero Angle Assumption: If and are the same ray, then mAEB = 0.
D) Straight Angle Assumption: If and are opposite rays, then mAEB = 180.
E) Angle Addition Assumption: If (except for point E) is in the interior of AEB, then mAEC + mCEB = mAEB.
Theorums
Linear Pair Theorum: If two angles form a linear pair, then they are supplementary.
Vertical Angle Theorum: If two angles are vertical angles, then they have equal measures.

The Unit Circle

When you work with angles in all four quadrants, the trig ratios for those angles are computed in terms of the values of x, y, and r, where r is the radius of the circle that corresponds to the hypotenuse of the right triangle for your angle.

Any two right triangles with the same base angle θ ("theta")will be similar in the technical sense of having their sides in proportion. This similarity is more obvious when when the triangles are nested:

Similarity (and thus proportionality) means that the trig ratios from the two nested triangles displayed above will be the same: Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved

The trig ratios for the same-sized angle θ are the same, even though the specific numbers from the two triangles' sets of sides are different. This emphasizes that, for trigonometric ratios, it's the angle θ that matters, not the particular triangle from which you obtained the angle.
To simplify computations, mathematicians like to fit an angle's triangle into a circle with radius r = 1. Once the hypotenuse has a fixed length of r = 1, then the trig ratios will depend only on x and y, since multiplying or dividing by r = 1 won't change anything. Only the values of x and y will matter.

The Unit Circle
The point of the unit circle is that it makes other parts of mathematics easier and neater. You can even now understand some of how the unit circle can be useful. For instance, in the unit circle, you have, for any angle θ, the trig values for sine and cosine are clearly sin(θ) = y and cos(θ) = x. Working from this, you can take the fact that the tangent is defined as being tan(θ) = y/x, and then substitute forx and y to easily prove that the value of tan(θ) also must be equal to the ratio sin(θ)/cos(θ).
Another thing you can see from this is that the values of sine and cosine will never be more than 1 or less than –1, since x and y never take on values outside of this interval. Also, since tangent involves dividing by x, and since x = 0 when you're one-fourth and three-fourths of the way around the circle (that is, when you're at 90° and at 270°), the tangent will not be defined for these angle measures.

Sine and Cosine
The sine (abbreviated "sin") and cosine ("cos") are the two most prominent trigonometric functions. All other trig functions can be expressed in terms of them. In fact, the sine and cosine functions are closely related and can be expressed in terms of each other.
Definition 1 is the simplest and most intuitive definition of the sine and cosine function. The sine definition basically says that, on a right triangle, the following measurements are related: * the measurement of one of the non-right angles () * the length of the side opposite to that angle * the length of the triangle's hypotenuse
Alternately, the cosine definition basically says that, on a right triangle, the following measurements are related: * the measurement of one of the non-right angles () * the length of the side adjacent to that angle * the length of the triangle's hypotenuse Futhermore, Definition I gives exact equations that describe each of these relations: sin() = opposite / hypotenuse cos() = adjacent / hypotenuse
This first equation says that if we evaluate the sine of that angle , we will get the exact same value as if we divided the length of the side opposite to that angle by the length of the triangle's hypotenuse. This second equation says that if we evaluate the cosine of that angle , we will get the exact same value as if we divided the length of the side adjacent to that angle by the length of the triangle's hypotenuse. These relations holds for any right triangle, regardless of size.
The main result is this: If we know the values of any two of the above quantities, we can use the above relation to mathematically derive the third quantity. For example, the sine function allows us to answer any of the following three questions:
"Given a right triangle, where the measurement of one of the non-right angles () is known and the length of the side opposite to that angle  is known, find the length of the triangle's hypotenuse."
"Given a right triangle, where the measurement of one of the non-right angles () is known and the length of the triangle's hypotenuse is known, find the length of the side oppositeto that angle ."
"Given a right triangle, where the length of the triangle's hypotenuse and the length of one of the triangle's other sides is known, find the measurement of the angle () opposite to that other side."
The cosine is similar, except that the adjacent side is used instead of the opposite side.
The functions takes the forms y = sin() and x = cos(). Usually,  is an angle measurement and x and y denotes lengths.
The sine and cosine functions, like all trig functions, evaluate differently depending on the units on , such as degrees, radians, or grads. For example, sin(90°) = 1, while sin(90)=0.89399....explaination
Both functions are trigonometric cofunctions of each other, in that function of the complementary angle, which is the "cofunction," is equal to the other function: sin(x) = cos(90°-x) and cos(x) = sin(90°-x).
Furthermore, sine and cosine are mutually orthogonal.

Trigonometric Identities
In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x =x" or usefully true, such as the Pythagorean Theorem's "a2 + b2 = c2" for right triangles. There are loads of trigonometric identities, but the following are the ones you're most likely to see and use.

Basic and Pythagorean Identities

Notice how a "co-(something)" trig ratio is always the reciprocal of some "non-co" ratio. You can use this fact to help you keep straight that cosecant goes with sine and secant goes with cosine. sin2(t) + cos2(t) = 1 tan2(t) + 1 = sec2(t) 1 + cot2(t) = csc2(t)
The above, because they involve squaring and the number 1, are the "Pythagorean" identities. You can see this clearly if you consider the unit circle, where sin(t) = y, cos(t) = x, and the hypotenuse is 1. sin(–t) = –sin(t) cos(–t) = cos(t) tan(–t) = –tan(t)
Notice in particular that sine and tangent are odd functions, while cosine is an even function.
Angle-Sum and -Difference Identities sin(α + β) = sin(α)cos(β) + cos(α)sin(β) sin(α – β) = sin(α)cos(β) – cos(α)sin(β) cos(α + β) = cos(α)cos(β) – sin(α)sin(β) cos(α – β) = cos(α)cos(β) + sin(α)sin(β)

Double-Angle Identities sin(2x) = 2sin(x)cos(x) cos(2x) = cos2(x) – sin2(x) = 1 – 2sin2(x) = 2cos2(x) – 1

Half-Angle Identities Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved

The above identities can be re-stated as: sin2(x) = ½[1 – cos(2x)] cos2(x) = ½[1 + cos(2x)]

Sum Identities

Product Identities

You will be using all of these identities, or nearly so, for proving other trig identities and for solving trig equations. However, if you're going on to study calculus, pay particular attention to the restated sine and cosine half-angle identities, because you'll be using them a lot in integral calculus.