Chapter 9 - Trigonometry - Preamble - Unit Circle and Angle Definitions

Oct 16, 2026

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Chapter 9 introduces trigonometry through an inverse-based perspective, shifting from function inverses over real numbers to a new relationship between angles and real outputs. The Unit Circle is defined as the radius-1 circle centered at (0,0), with constraints on coordinates x and y and an emphasis on the identity x^2 + y^2 = 1. It then defines angles as counterclockwise circular distance from the starting point (1,0) to a terminal point t, connecting angle measures to degrees and discussing why 360 is used.

LanguageEnglish

CategoryExam

FormatPDF

Pages27

Reading Time1 h 8 min

AccessFree to download

Preamble

Inverse approach to trigonometry

The Unit Circle

Defining the Angle on the Unit Circle

Chapter 9 Trigonometry Preamble Why introduce trigonometry when this is a text about College Algebra? Aren’t we supposed to wait until the next class, usually named “Precalculus” or “College Algebra and Trigonometry” in order to talk about sines and cosines, triangles, and so on? If you look at the title of the book, I call it an“inverse approach”. That means that I’m heavy into inverses, and of all kinds. This study we undertake in this chapter is somewhat different, in that up to now, our domains and ranges of our functions and their inverses have all been the same creature; they have all been real numbers. Now, we are going to take angles and our function will turn them into real numbers; the inverse will take these real numbers and turn them into angles. It’s kind of like if our domain was cars; vehicles with four wheels, and up until now our functions have changed them somewhat, but they are still cars. For example, we have a “yellow” function in which all the cars that go through this function are painted yellow. Our domain for this function is any color car with four wheels; our range for this function is yellow cars with four wheels. Another function in our example here would be to have the domain be cars with stock engines, and our function would be to add a turbocharger and exhaust headers to improve performance. The range would be cars that were much more powerful; they would look the same but have better performance. The inverse of this function would remove the turbo and headers and convert the car back to its original performance standard. But in this chapter, we will be like taking our domain of cars, and our function turns them into rabbits. The inverse of this function would take rabbits, and turn them into cars. The Unit Circle We start by defining the Unit Circle. This is simply a circle on the coordinate plane whose radius is 1 and whose center is at the origin, (0,0). Let’s talk about what we know about points on this circle. Because the radius is 1, we know that every point on the circle is exactly 1 unit away from the origin; that is, the distance from the origin to the point is exactly 1. We also know that every point on the circle is described by a coordinate pair (x , y). We know from our distance formula, then, that (x − 0)2 + (y − 0)2 = 1 What I want to now stress is that not just any x and any y will work here; we now have some constraints on x and y. Another way to say it is that if we know what y is, we have a very limited choice on what x can be; if we know what x is, we have a very limited choice on what y can be. Here is a list of constraints on x and y: 1. −1 ≤ x ≤ 1 2. −1 ≤ y ≤ 1 3. y = ± 1 − x 2 (If we know what x is, these are our two choices for y) 4. x = ± 1 − y 2 (If we know what y is, these are our two choices for x) 5. x 2 + y2 = 1 (This last one is a tool that will be frequently pulled out of your toolbox when you deal with trig functions!) Now, let’s start finding some points on this circle, using the above tools. Suppose that we pick x = 2 . Whoops, that won’t work; it fails the first constraint. Alright then, let’s put the works. 2 in the denominator, and pick x = 1 2 . That Then, what is y? Constraint \#3 says to make which equals y = ± 1 − ( ~~ ~~12~~ ~~ )2 ± 1 − ~~ ~~14 which equals ± or ± ~~ ~~2~~ ~~ or approximately ±0.866 On the unit circle, then, we see these two points: Do you notice something here? Something that we first talked about when we defined functions? We have one x-value going to TWO y-values. Hey! This doesn’tobey the Vertical Line Test. You will find in the work ahead that we will somehow constrain our domain so that we only have x going to exactly one y-value, because we are interested in functions here. The big takeaway here is that we chose x, and by doing so immediately limited what y could be. The points HAVE to lie on the circle. Had we chosen y, we would immediately be limited to wha

FAQs

Why does the chapter introduce trigonometry at this point in College Algebra?

It explains that the chapter extends inverses from real-number domains and ranges to angles mapped to real numbers, and then inverted back to angles.

What is the Unit Circle in the chapter’s definition?

It is a circle on the coordinate plane with radius 1 centered at the origin (0,0), so every point on the circle is exactly 1 unit from the origin.

How is an angle defined on the Unit Circle?

An angle is the distance traveled along the circle C around its center O in a counterclockwise direction from the starting point (1,0), labeled 0, to a terminal point t.

Oct 16, 2026

Upload by OmBimo