When I began this endeavour, I thought to myself "You know, I'd sure like to learn calculus." Like you do. I had a sneaky suspicion I would need to get a handle on trig first, but I'm rather steadfast when I set my mind to something, so I wanted to give it a shot anyway.
One trip to my local used bookstore later I came home with the gem you see to the left, promising to deliver "Calculus Demystified" to the uninitiated such as myself. I gather this is sort of similar to the "for dummies" series that has been so popular - I've never read one of those, and honestly I had my reservations about this one over a more traditional text, as well, but I picked it for several reasons. One, I had enough trade credit at this particular shop to get it for absolutely free, and two, it offered an introductory chapter which included reviews of geometry and trigonometry before getting into calculus. I thought it would be a good litmus test of my abilities to work through this introductory material, and then decide how to proceed.
It began simply enough, with some remarks on how the calculus is fundamental to all of modern science, explaining the difference between derivative and integral calculus, and then jumps right in by laying out the different types of numbers: natural, integers, real, rational, irrational (and imaginary, which didn't appear there), and then some linear and planar geometry exercises. I was feeling pretty good about myself as I was halfway through the first chapter and hadn't needed to so much as put pencil to graph paper. Slope-intercept form came back to me like a welcome old memory, and before I knew it I was graphing loci of equations with relative ease - and then I got to the section on trigonometry.
Now, I'm sure that this chapter, to people who have a familiarity with the principles of trig and are simply seeking a review, would be a valuable and concise resource. To me, the uninitiated, it lost me. Now - I wasn't disappointed, in fact I'd sort of expected this - now, at least, I knew exactly where I stood. My algebra and geometry were pretty solid, I just needed to start with trigonometry for now, and tackle calculus only after I have a strong handle on trig.
SOH-CAH-TOA
One of the reasons I'm keeping this blog is because I don't want to be "tied" to a single text or resource on these subjects. By chronicling my experience online I am also welcoming suggestions and insights from my readers and followers, and I've not been disappointed. For instance - I told a friend that I was going to be taking on this project and she immediately responded "Oh neat! Do you know about Khan Academy?" I didn't, but I am certainly glad that I do now.
Khan Academy is a learning site featuring training and exercises on a wide variety of subjects, and today I went through the "Basic Trigonometry" module and some practice questions. This resource, unlike my admittedly more advanced and less appropriate used-bookstore find, started at the beginning and introduced me to the basics of trigonometry, with which I had a passing familiarity, with refreshing clarity. It also introduced me to an acronym / mnemonic device that I had forgotten I had even heard before: SOH-CAH-TOA. This was a revelation.
For those among you who are not math savvy, I'll explain:
One trip to my local used bookstore later I came home with the gem you see to the left, promising to deliver "Calculus Demystified" to the uninitiated such as myself. I gather this is sort of similar to the "for dummies" series that has been so popular - I've never read one of those, and honestly I had my reservations about this one over a more traditional text, as well, but I picked it for several reasons. One, I had enough trade credit at this particular shop to get it for absolutely free, and two, it offered an introductory chapter which included reviews of geometry and trigonometry before getting into calculus. I thought it would be a good litmus test of my abilities to work through this introductory material, and then decide how to proceed.
It began simply enough, with some remarks on how the calculus is fundamental to all of modern science, explaining the difference between derivative and integral calculus, and then jumps right in by laying out the different types of numbers: natural, integers, real, rational, irrational (and imaginary, which didn't appear there), and then some linear and planar geometry exercises. I was feeling pretty good about myself as I was halfway through the first chapter and hadn't needed to so much as put pencil to graph paper. Slope-intercept form came back to me like a welcome old memory, and before I knew it I was graphing loci of equations with relative ease - and then I got to the section on trigonometry.
Now, I'm sure that this chapter, to people who have a familiarity with the principles of trig and are simply seeking a review, would be a valuable and concise resource. To me, the uninitiated, it lost me. Now - I wasn't disappointed, in fact I'd sort of expected this - now, at least, I knew exactly where I stood. My algebra and geometry were pretty solid, I just needed to start with trigonometry for now, and tackle calculus only after I have a strong handle on trig.
SOH-CAH-TOA
One of the reasons I'm keeping this blog is because I don't want to be "tied" to a single text or resource on these subjects. By chronicling my experience online I am also welcoming suggestions and insights from my readers and followers, and I've not been disappointed. For instance - I told a friend that I was going to be taking on this project and she immediately responded "Oh neat! Do you know about Khan Academy?" I didn't, but I am certainly glad that I do now.
Khan Academy is a learning site featuring training and exercises on a wide variety of subjects, and today I went through the "Basic Trigonometry" module and some practice questions. This resource, unlike my admittedly more advanced and less appropriate used-bookstore find, started at the beginning and introduced me to the basics of trigonometry, with which I had a passing familiarity, with refreshing clarity. It also introduced me to an acronym / mnemonic device that I had forgotten I had even heard before: SOH-CAH-TOA. This was a revelation.
For those among you who are not math savvy, I'll explain:
To the right is a triangle. In particular, it's a right-triangle. The little box in the corner tells us that angle is 90-degrees. Trigonometry, essentially, is the study of right triangles and the relation between their dimensions and angles. It's used largely in astronomy, which is a big reason why it interests me. So - if you have ever walked past a classroom where trig was being taught, you may be familiar with the basic functions: sine (or sin), cosine (cos), and tangent (tan). These are the functions that explain how triangles work, and SOH-CAH-TOA is essentially there to help us remember how to calculate them.
SOH stands for sin=opposite/hypotenuse, CAH stands for cos=adjacent/hypotenuse and TOA of course stands for tan=opposite/adjacent.
For example in the 3-4-5 triangle here, let's say we wanted to calculate these functions for the angle highlighted in blue, designated theta (Θ). Here's a note. In trigonometry, the angle in question is usually represented by Θ. As far as I can tell, you can think of it as the ubiquitous "x" of Algebra, simply a variable - I presume its consistent use in trigonometric functions like this only serves to help keep notation consistent so we can more easily recognize what kind of math we're doing when we glance at an equation.
So if we wanted to calculate the sine function of Θ, we simply look at our mnemonic device and plunk in the numbers. In this case, sinΘ=3/5. "Opposite" means the side opposite the angle in question, and hypotenuse is of course the long side.
It then follows that cosΘ=4/5, as the side measuring 4 units is adjacent to Θ, and finally, tanΘ=3/4.
Rad - and not the ninja turtle kind.
Now, a big learning curve, for me, in trigonometry as opposed to geometry is that angles in trig are not calculated in degrees, but in radians. A full circle, 360 degrees, is expressed in radians as 2π (that's 2 pi), and the 90 degree angle in our triangle above would be expressed as π/2 (since it's a quarter of a circle, and 1/4 of 2π is π/2). Now - my understanding of this part is still in its early stages, admittedly, but I'll be diving into this soon enough as I continue learning and - before someone asks, yes I do plan to devote a future post to the ever-popular "Pi is Wrong" debate. I'd read the Tau Manifesto before, and found it intriguing, but of course I had that barrier of not really understanding the full implications... I read it again after just preliminary trigonometry, and I think I'm starting to see the beauty of τ.
Stay tuned for the unfolding saga as I continue my own "Trigonometry I" course, including more on π, τ, and other meditations on irrational numbers, which are completely fascinating to me.
And by the way, I am keeping my copy of Calculus Demystified and will be referring to it more as I grow into it. It's just a little advanced for me yet.
SOH stands for sin=opposite/hypotenuse, CAH stands for cos=adjacent/hypotenuse and TOA of course stands for tan=opposite/adjacent.
For example in the 3-4-5 triangle here, let's say we wanted to calculate these functions for the angle highlighted in blue, designated theta (Θ). Here's a note. In trigonometry, the angle in question is usually represented by Θ. As far as I can tell, you can think of it as the ubiquitous "x" of Algebra, simply a variable - I presume its consistent use in trigonometric functions like this only serves to help keep notation consistent so we can more easily recognize what kind of math we're doing when we glance at an equation.
So if we wanted to calculate the sine function of Θ, we simply look at our mnemonic device and plunk in the numbers. In this case, sinΘ=3/5. "Opposite" means the side opposite the angle in question, and hypotenuse is of course the long side.
It then follows that cosΘ=4/5, as the side measuring 4 units is adjacent to Θ, and finally, tanΘ=3/4.
Rad - and not the ninja turtle kind.
Now, a big learning curve, for me, in trigonometry as opposed to geometry is that angles in trig are not calculated in degrees, but in radians. A full circle, 360 degrees, is expressed in radians as 2π (that's 2 pi), and the 90 degree angle in our triangle above would be expressed as π/2 (since it's a quarter of a circle, and 1/4 of 2π is π/2). Now - my understanding of this part is still in its early stages, admittedly, but I'll be diving into this soon enough as I continue learning and - before someone asks, yes I do plan to devote a future post to the ever-popular "Pi is Wrong" debate. I'd read the Tau Manifesto before, and found it intriguing, but of course I had that barrier of not really understanding the full implications... I read it again after just preliminary trigonometry, and I think I'm starting to see the beauty of τ.
Stay tuned for the unfolding saga as I continue my own "Trigonometry I" course, including more on π, τ, and other meditations on irrational numbers, which are completely fascinating to me.
And by the way, I am keeping my copy of Calculus Demystified and will be referring to it more as I grow into it. It's just a little advanced for me yet.
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