Geometry How Can We Improve a Free Trigonometry Textbook for High School Students?

[josephgerth]

Good morning everyone,

I have written a free math textbook, and I'd appreciate some feedback on it. It's about the basics of Trigonometry, including sine, cosine, tangent, radians, the unit circle, a bit on identities, and the Law of Sines, Cosines, and Tangents. I wrote it in a rigorous manner, attempting to be very formal, avoid gimmicks, avoid mere recipes, and share much of the theory and intuition behind these fundamental concepts in Trig.

I wrote the book for 9th/10th grade Geometry students. I had a few goals with this text: First, I had a good class, and I wanted to give them a reasonable. Second, Trig is sorely lacking in our curriculum, and I found that students were completely unprepared for Trig in later classes, and as a result, many struggled in subsequent classes, particularly in Pre-Calculus. Finally, I wanted to give my students a formal math experience, in hopes that they might find it enjoyable - or, at least, more enjoyable than the contemporary math books they've used up to this point.

As you can see, this is the 1st Edition, and I want to make some improvements for the 2nd edition.

Finally, this text is completely free. You may copy, print, share, etc. this text as you desire.

Thank you for your time and your feedback.
Joe

 

[micromass]

Looks nice. I have some comments though:

1) I would like to have seen the converse of the Pythagorean theorem mentioned, because this shows that the Pythagorean theorem connects algebra with geometry in a really neat fashion
2) I would like to see more applications. You can't really understand trigonometry without them, and it really is where the subject shines. For example, show how you can calculate the radius of the earth, the height of a building or the speed of a ship using trigonometry
3) I would like to have seen a picture of the trigonometric functions in one diagram such as this:

but then with the cotangent, secant and cosecant also added. The nice thing is that you only need to memorize this picture and you can just "see" a lot of trionometric identities. For example, stuff like $\sin(x) = \sin(\pi - x)$ is something you can just see, and it doesn't need to be memorized anymore. Additionally, it partially explains where the name tangent comes from.
4) Is there any reason to be so brief on the identity part of the text? You didn't mention cool stuff like the double-angle or half-angle formulas. These are really crucial later on too. Furthermore, it is not often mentioned in high school books, but the tangent half-angle formulas ( https://en.wikipedia.org/wiki/Tangent_half-angle_formula ), make 90% of all tricky trigonometry questions trivial since it reduces possibly complicated expressions involving cos, sin, tan, cotan, sec, cosec, etc. to an easy equality of polynomials. I still want to see a high school text having the guts to mention this (and thereby making it harder to find good test questions). It does give a neat way to let a computer check a trigonometric identity though. And while we're on the topic, it would be cool to show how these half-angle formulas generate all Pythagorean triples.

 

[MidgetDwarf]

Geometric proof the basic trig identities. Ussually done with 2 diagrams. It includes to 2 applications of the distance formula to get law of cosines. From the second geometric diagram, we can get most of well used trig identies. Including the often forgot en sin a - sinb formula.

 

[josephgerth]

Looks nice. I have some comments though:

1) I would like to have seen the converse of the Pythagorean theorem mentioned, because this shows that the Pythagorean theorem connects algebra with geometry in a really neat fashion
2) I would like to see more applications. You can't really understand trigonometry without them, and it really is where the subject shines. For example, show how you can calculate the radius of the earth, the height of a building or the speed of a ship using trigonometry
3) I would like to have seen a picture of the trigonometric functions in one diagram such as this:
but then with the cotangent, secant and cosecant also added. The nice thing is that you only need to memorize this picture and you can just "see" a lot of trigonometric identities. For example, stuff like $\sin(x) = \sin(\pi - x)$ is something you can just see, and it doesn't need to be memorized anymore. Additionally, it partially explains where the name tangent comes from.
4) Is there any reason to be so brief on the identity part of the text? You didn't mention cool stuff like the double-angle or half-angle formulas. These are really crucial later on too. Furthermore, it is not often mentioned in high school books, but the tangent half-angle formulas ( https://en.wikipedia.org/wiki/Tangent_half-angle_formula ), make 90% of all tricky trigonometry questions trivial since it reduces possibly complicated expressions involving cos, sin, tan, cotan, sec, cosec, etc. to an easy equality of polynomials. I still want to see a high school text having the guts to mention this (and thereby making it harder to find good test questions). It does give a neat way to let a computer check a trigonometric identity though. And while we're on the topic, it would be cool to show how these half-angle formulas generate all Pythagorean triples.

Thanks for the reply and the feedback! A few replies from me...

1.) That's a great suggestion. I hint at this idea subtly in the exercises, but it certainly could use some more attention. It also wouldn't add that much tangential (no pun intended) material.

2.) Agreed on this point. My thinking is that students will get plenty of time to work application problems in later courses (and the science curriculum uses basic Trig functions, too) whereas students rarely get any rigorous, theoretical treatment of math topics. But I completely see your point, and although I have a few application problems, some more would be beneficial.

3.) This is an excellent suggestion. Thank you. I do use a similar picture in the exercises for section 3.2 to show the identities $$tan^{2}{\alpha}+1=sec^{2}{\alpha}$$ and $$cot^{2}{\alpha}+1=csc^{2}{\alpha},$$ although it's never really dug into. Definitely something I'll be adding.

4.) An excellent question. Because of the intended audience, I chose to go light on the identities. Certainly we could spend more time on this (worthwhile) topic, but given that this is a course intended for 9th/10th graders, I thought that I had enough. Perhaps an honors course could use more. Additionally, this is by no means the last time they'll see identities (they'll see them at least once more and in far greater detail in Pre-Calculus). As such, I chose to only include the identities that were very easy to prove (namely, the Pythagorean Identities).

As I said, I really appreciate the feedback! I'm hoping to hear from some more folks so I can make the 2nd edition that much better.
Joe