d4 quick review of topics in trigonometry trigonometric functions equations quick review notes ebooks quick review of topics in trigonometry trigonometric.
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Thanks Doug, really glad it helped. Math and any subject, really can be so much easier to learn when we look for an approach that gets things to click deep down. Khalid , Trigonometry is used to be Tricknometry but now I find it is as simple as eating banana. Great job well done. I love your creative thinking. For the percentage you can also use this app: It gives you something like that: For more playing around: I never realized it was this simple. Hi Khalid, Keep helping the world in this Wonderful way. Thank you very much. Always waiting for ur new ideas in simplifying the concept.
However, when you visualise the Tan function in the 3rd Quadrant, intuitively it feels like it should be negative. Added complication is that if you take the word TANGENT literally as a slope of the circle then at 90 degrees the slope of the tangent should be zero. But Tan function is undefined at 90 degrees division by zero at this point. How can I visualize this better?
I do think most ideas can be as simple as falling off a log if seen the right way. Hah, you should have seen the original post, which was about twice as big: I was in the same boat, thinking I had to memorize everything. Article was getting big, something good for the follow-up! Using the percentage analogy, tangent is the height relative to the wall distance, but each component can have a sign:. So this is negative height on the back wall, which counts as positive: Actually I am doing the same that you are doing, breaking everything down, not taking formula as-is, trying to find the insights behind… but just in German and a tiny bit more animated ;-.
Some English speakers have asked me already to transfer my videos into English. I think I will give it a try this year, if I find time. I will send you a message as soon as the first video is ready. While I could follow the explanations, I did want to follow your advice and not get too hung up on an individual diagram. I also wanted to play around with the concepts, so I put together the following demos on the online Desmos calculator:.
Putting these demos together and seeing the results also helped make everything clearer, and I thought others might find these useful. I saw this last night before going to bed, and used it this morning with my Geometry class as we began our Trig unit today. After reading it last night, presenting it to the kids this morning, and reading through this again, trigonometry finally makes intuitive sense to me. I am confident that this will help my students see this in a clearer light, and hopefully the handout that I put together to introduce sine and cosine today is helping them make meaningful connections.
I am a middle-aged high school English teacher who does math as a hobby in the mornings. I am fascinated with trigonometry and just stumbled on your site yesterday. I really appreciate the visual analogies.
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I have one question and it is not that important, but it is nagging at me. In your reference to the Vitruvian Man you mention that you can also deduce the eye width with the information given. I could not figure this out. Thanks Eric, glad you enjoyed it. Good question — the eye width trick was something I remembered from drawing comic book characters when young.
When that radius is extending to the 3rd quadrant, the slope remains the same sign and size. Kalid, you have a beautiful way of explaining things. Your illustrations, intuition buildup and Aha! Everything just falls into place never to be forgotten again. I am confused about the ceiling diagram. How come height traversed is always 1? Seems height can be bigger as the line extends beyond the dome.
Hi vinay, try this interactive calculator for an example: When building a ramp up to the ceiling, the distance we travel depends on the angle we pick. However, the ceiling itself is always 1 unit above the ground. In a building the ceiling is always a constant height, no matter how steep the stairs are to get there. Dear friend, Thank you for this precious point of view about trigonometry. It was a pleasure reading your article. My apologies for the delay on approving that comment, it was stuck in my moderation queue because of an overactive spam filtering rule.
First off…I am completely in agreement with Alecks on the insight. Kalid, your website has added an immense amount of intuition to my understanding of mathematics. Thank you for your fresh approach to the topics you cover. After reading this I began getting into hyperbolic trig functions. I think, since you love e so much, you could provide a lot of intuition on these functions since their definitions involve 2 terms of e.
Thanks for your time. Also, we can even define the regular trig functions in terms of e as well: Somehow last night I went from the triangles in the circles to the wedges formed by the secant and tangent lines. That is the beauty of the insights you provide they build up our own abilities to make new connections. Kalid if u remember I messaged you regarding trigzz!! U re genius indeed.
But your dome analogy is far more efficient and natural. Thanks Harish, really glad it helped: It was only recently i. Also the fact that the unknown sides are percentages of the known sides is seriously illuminating. You could shorten the explanation by cutting some of the anatomy content as well as the higher Trig references. But well done for this explanation which I am unlikely to forget anytime soon.
Thanks Alan, glad the analogies helped. The anatomy part helps me realize the role of trig way to explore an alien shape but everyone has a different takeaway: I have been battling this issue with teachers for years now. I agree, memorizing acronyms is a poor substitute for internalizing the actual concept they should serve as reminders, not lessons.
I honestly do not know of another site dedicated to teaching the underlying concepts as a means to understand the topic overall. You have been a godsend for me in math. Hi Kalid, Once again, thank you for helping everyone see how the trig functions can be applied in real life. Perhaps you can share some insight as to how you came about the two above examples. Technically, sec and cosec have a minimum distance of 1, so can take any value from 1 to infinity, or -1 to -infinity when facing backwards.
I have a general article on my strategy http: I thought of a dome after too many IMAX movies maybe? I hope to write more about this too. Excellent article, thank you VERY much for sharing. My life would be so much easier in school if teachers made these relations clear. So, the hypotenuse is the secant or the radius? Good question on the meaning of the hypotenuse. In this case, the secant is written in terms of the hypotenuse, such as 2. This is also the radius of the dome.
Ok, now I got it! Thank You Kalid Binnoy http: You want to rewrite the science of Trigonometry? Anonymous or what do you call yourself, we know you are a genius but keep it to yourself. We are satisfied with what Khalid is givining. If you so much know trig as you claimed, why have we not heard about you uptil now? Are you telling me that that the traditional teaching style has helped maths? Now, someone comes up to help majority of us, you are not included, you are scathingly criticising.
Please, go and sit down. I keep aside a few hours every week for exploring math, and I read this article last week during that time. Now, today, during one of those hours, I was amazed to find that the explanations were still clear in my brain. Hi Kalid, Longtime lurker of your site, admirer of your ability to intuit. I appreciate examples like the hellish voyage on the appendix. I found your site way after leaving college and things are finally becoming clear to me. Thanks for your passion.
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My big slide rule has trig functions and exponential functions in the form of number lines that refer to the basic log scale. I think my generation and those before us had a better visual and intuitive grasp of math because of the slide rule. Almost all calculations on a slide rule are based on proportions. They become natural if you work with a slide rule.
Motivation: Trig Is Anatomy
A slide rule user must always know where they are in a calculation. You might make little mistakes in the 3rd or 4th significant digit, but never a BIG electronic calculator type of mistake. I have never stopped using a slide rule and never really graduated to electronic calculators.
Their are lots of them floating around and the web has lots of info on how to use them. Having a tangible representation of what logs are doing is another way to build an intuition, might be a fun article! 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 years of development and evolution. And they were expansive. Both are in working order today after years. Back then, a person bought usually one high quality one in a lifetime.
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. The result pops out magically. They are highly anti-intuitive.
This helps a lot! Calculus was way easier, but trig popped up everywhere in it and in physics too and I was just at its mercy! Trig bugged me for so long as well until I found a way to have the relationships click. Salam wa alay kum …. Khalid and very very thanx for providing such sites …I think this is the best site for math seekers….
This posting is great! Makes it a lot easier to understand. This certainly made it easier. 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 could always work with the formulas, but had a hard time visualizing them and understanding them intuitively.
This really helped, thank you! 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! One question though, is there a reason why the tangent is always vertical??
Trigonometry - Wikipedia
See the section of the article Appendix: The Original Definition Of Tangent. Hi Ayubi, glad you enjoyed it! It might be a fun article, but I essentially look for connections between things wherever I can. 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. 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.
Kudos, Kalid, for making this subject so easy and understandable! If only all high school teachers everywhere would watch your videos. Of all my years in sports medicine and direct research in human performance, I have never heard of a fibia. I am assuming you either meant tibia or fibula.
Thanks for the laugh. The ceiling and wall are very similar. Thank you so much for this Kalid! You are absolutely amazing!! I am starting to enjoy maths so much more after reading your articles. You are the best guy and teacher I have ever seen in my life, nobody has explained to me that well in my life , aha moment …. Kalid thanks a ton. Thought these math gifs might help someone else here. Thank you so much! I thought that I was the only one that recognized trig functions as percentages.
However, I never came up with a cool analogy to remember their relationships, thanks for the article: My math teacher just sent this to me and it helped me understand trig better.
Introduction to Trigonometry
Why is secant of degrees a negative value but tangent of degrees a positive value? I have used the desmos visualization and I still cannot get it. I have been practicing the quizes at http: Surely if the tangent becomes positive then the hypotenuse to that tangent should be positive and if the tangent is negative then the hypotenuse to that tangent should be negative? How does math help us in life??? Algebra 2 Equations and inequalities Overview Solve equations and simplify expressions Line plots and stem-and-leaf plots Absolute value Solve inequalities. Algebra 2 How to graph functions and linear equations Overview Functions and linear equations Graph functions and relations Graph inequalities.
Algebra 2 How to solve system of linear equations Overview Solving systems of equations in two variables Solving systems of equations in three variables. Algebra 2 Matrices Overview Basic information about matrices How to operate with matrices Determinants Using matrices when solving system of equations. Algebra 2 Polynomials and radical expressions Overview Simplify expressions Polynomials Factoring polynomials Solving radical equations Complex numbers. Algebra 2 Quadratic functions and inequalities Overview How to graph quadratic functions How to solve quadratic equations The Quadratic formula Standard deviation and normal distribution.
Algebra 2 Conic Sections Overview Distance between two points and the midpoint Equations of conic sections. Algebra 2 Polynomial functions Overview Basic knowledge of polynomial functions Remainder and factor theorems Roots and zeros Descartes' rule of sign Composition of functions. Algebra 2 Rational expressions Overview Variation Operate on rational expressions. Algebra 2 Exponential and logarithmic functions Overview Exponential functions Logarithm and logarithm functions Logarithm property. Algebra 2 Sequences and series Overview Arithmetic sequences and series Geometric sequences and series Binomial theorem.