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You have to observe that $$Opposite = \frac{Opposite}{Adjacent}*Adjacent = \frac{Opposite}{Hypothenuse}*\frac{Hypothenuse}{Adjacent}*Adjacent = \frac{\sin(\theta)}{\cos(\theta)} *Adjacent= \tan(\theta)*Adjacent$$. So you start with an unsual math-trick and multiply by one. Now you can concentrate on $\frac{Opposite}{Adjacent} =\frac{\sin(\theta)}{\cos(\... 0 Like the other answers, I'd also suggest lots of computations for many triangles of increasing steepness. This way students could gradually figure out what the curves are like, what the domains and ranges are, and where it increases, decreases, etc. They could sketch approximations of the functions? So, for example with$\tan(62)$, even if they don't know ... 0 Another mnemonic attempt (despite not having much sense): The cones in my pie are squared, because their age is over three. Getting together, we have that the volume of a cone is$\pi r^2\cdot \dfrac h3$. 1 I'm not sure if I would class this as a mnemonic, except in the most basic definition of "a way to remember": A cone is basically a three-dimensional triangle. Triangles have an area of$\frac12bh$; but triangles are two-dimensional, so it makes sense that a three-dimensional version would use$3$instead of$2$, yielding$\frac13bh$. The "base" of a cone ... 1 The comments give great suggestions for how to understand and remember the formula, but since you asked for a mnemonic specifically, how's this? Henry won third in the creative desserts contest because his pie are squared.$h\cdot (1/3) \pi r^2$Edit: someone "fixed" this by changing "are" to "is." The mnemonic falls apart if you do that, so please leave ... 2 The common way to introduce how trig functions are calculated numerically is via Taylor/Maclaurin series. These are used extensively in a lot of engineering and physics based applications. However, this requires a knowledge of calculus, and isn't very useful in the context of basic trigonometry. It also isn't useful for giving students a "feel" of how these ... 0 There are a lot of approaches to approximating sine and cosine which would be sensible to a high school audience. One method is to use the double angle formula repeatedly, together with small angle approximations. For instance, if you want to approximate$\sin(\theta)$and$\cos(\theta)$, approximate$\sin(\theta/64) \approx \theta/64$and$\cos(\theta/64)...

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If you are teaching this at an introductory level, then the algorithm that calculators use today is going to go far over their heads. (It might go over MY head!) The story of how we developed increasingly accurate trig tables over the course of history would be an interesting topic of inquiry for advanced algebra / pre-calculus / calculus, but at the ...

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Expanding on the point already mentioned by guest's answer, I think you should stop focussing on building intuition. Instead, focus your time on solving the exercises even if, by the time you solve them, you don't feel like you've gained much in terms of intuition. Just solve the problem and move on. This might seem very counterintuitive, but the point is ...

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Regarding retention, doing lots of problems can help with this. Also, asking lots of questions about what you’re reading in the book, and trying to figure them out. You mentioned field extensions, so one question might be: “It’s a bit counterintuitive that this book spends more time talking about field extensions than about fields themselves. What’s so ...

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I agree that insight sometimes takes time, but don't dismiss the benefit of exercises. Sometimes, the insight monster will lurk in the background in your subconscious. If you do exercises, it may come out and give you the revelations later, not directly from the drill, but somehow aided by the time/exposure. If you solely chase insight and don't build ...

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