One might get a taste by using an online simulation, but there is no substitute for holding one your hands. They are precision instruments, the final product of 300 years of development and evolution. And they were expansive. My high school slide rule cost $55 in 2014 dollars, and my big scientific-engineering one was $250. Both are in working order today after 55-60 years. Back then, a person bought usually one high quality one in a lifetime. The German-American rocket engineer Werner von Braun worked on the Apollo moon shots with one he bought in the early 1920’s.
(I have never replaced a set of batteries…)
I am not suggesting one really learns to use one seriously. With all the relations there in front of you, they are great objects for meditations on mathematics. Most of your insights in this site could be arrived at by figuring out precisely how a slide rule works. If a person uses a slide rule a lot, you can estimate the result of a computation by closing your eyes, picturing a slide rule, and operating it in your imagination. Calculators are of course remarkable, but they are “black boxes”. Nobody knows what’s going on inside. The result pops out magically. They are highly anti-intuitive.
Get your hands on a physical slide rule. Beware: they cast a spell.
I am working my way through your entire site. Great stuff, new little things at every point.
Thank you! This helps a lot! I am on winter break, and am studying math on my own to " understand" it instead of just memorizing it. I’ve had two physics classes and up to Calculus I but I am very weak at Trig because I have no idea, it just doesn’t click. Calculus was way easier, but trig popped up everywhere in it and in physics too and I was just at its mercy! The diagram was the best… I always wondered where the other identies were located on the triangles/ circle dome. the most useful diagram next to the original unit circle I’ve ever come across!
Hi.
This posting is great! Makes it a lot easier to understand. I knew how to do Trigonometry but realized just how tricky it was when I had to try & explain it to my teenage son. This certainly made it easier. Thank you.
I just had an awesome aha moment that I wanted to share: another reason why radians are so useful when using trig is that since sine and cosine are in the form of percentages with respect to the radius, by using radians (which are in terms of radius units) we actually are already on the unit circle, since we divide the distance traveled by r.
Just a small tip for anyone out there using Google as a calculator for any of these functions, beware that it returns results in radians. I was puzzled for a few minutes on Kalid’s example to solve for sine of angle x until I realised the answer Google was computing for arcsin(.60) [0.64] was in radians. Performing a conversion from that to degrees (by multiplying the radian value by 180/π) yielded the expected result of 36.9 degrees. Happy Math!
Wow…I feel like my foot has been itching since high school, and it took me 18 years until I was finally able to take off my shoe and scratch it. I can’t believe how much sense this post makes. Kudos, Kalid, for making this subject so easy and understandable! If only all high school teachers everywhere would watch your videos.
Hey Khalid, it’s really nice to see how you demystify many of the concepts that we learned in high school. I’ve always tried to get some intuition but failed so far so I gave up on this quest. It’s refreshing to see how you’ve been able to make it work. However, I would love to see how you reasoned your way through to getting these 3 similar triangles all stacked up on top of one another. Can you please post an article or a reply to this comment? Would love to see more of your work!
Hi Ayubi, glad you enjoyed it! It might be a fun article, but I essentially look for connections between things wherever I can. All the trig functions seem to have similar relationships (something^2 + something^2 = something^2) which fits the Pythagorean model. Seeing everything as percentages seemed to help clarify as well (tan is just another percentage, except it can go to infinity). A lot of it is trial and error and a belief that things can be simple if we look at it the right way (and it may take a lot of time before it jumps out at us).
Hi Geraint, the tangent doesn’t have to be vertical, but it’s lined up that way for simplicity in the diagram. See the section of the article Appendix: The Original Definition Of Tangent.