I want to know what is the best way to teach simple trigonometric equations, such as $$\sin x=0$$. Should I use the trigonometric circle or the sine graph? Which is better?
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4$\begingroup$ Tough to answer this with no context. Solving this type of equation is usually part of a curriculum where both the graph and unit circle have already been introduced. Who is it you are teaching? $\endgroup$– JTP - Apologise to MonicaDec 9, 2015 at 13:12
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$\begingroup$ Yes, of course unit circle and the graph, have been introduced. It is at 10th grade in my country. $\endgroup$– amarius8312Dec 9, 2015 at 14:45
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1$\begingroup$ Are there any standards that you are supposed to teach to? What do the standards say? $\endgroup$– celerikoDec 9, 2015 at 14:47
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1$\begingroup$ You might wish to check out Jenna Van Sickle's doctoral dissertation here: A History of Trigonometry Education in the United States: 1776-1900. $\endgroup$– Benjamin DickmanDec 10, 2015 at 22:23
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$\begingroup$ "Which is better?" Why do you assume that one way is "better" than the other? $\endgroup$– John ColemanJul 16, 2016 at 2:34
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3 Answers
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Since the students have already been exposed to the unit circle and the graph, I would give them the equation and ask for discussion of how to figure it out. If the students come up with the answers themselves they are most likely to remember it far better than if you tell them.
If they only come up with one way, present the other way as well. If they have no idea, present both ways as tools and ask which would help.
I believe self-discovery is best.
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Start with the unit circle to give them an intuition for what it means (& see this answer for some diagrams that might help). Then teach students how to plot a function from input-output pairs (if they don't know that already), and put that skill to work so that they are deriving the graph from the intuition. This will permit a deeper understanding than presenting the graph first, and help students be able to figure out key points (like, "does this one start at 0 or 1?") when they've forgotten memorized graph specifics but retained the original intuition of where the curves comes from.
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Start with triangular approach first , and then go to unit circle approach
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3$\begingroup$ Can you explain why one should do it this way? $\endgroup$ Dec 10, 2015 at 12:46
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$\begingroup$ The right triangle approach to cosine and sine is not adequate as a definition, because it only makes sense for angles between zero and $\pi/2$. Looking at the unit circle, it's clear for which angles the $y$-coordinate is zero. Looking at right triangles... what would a "triangle with side length zero" even mean? One can make sense of the idea, but it'd introduce a lot of needless confusion, and it's even worse for angles larger than $\pi/2$. $\endgroup$ Dec 10, 2015 at 15:13
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2$\begingroup$ -1 for lack of explanation and detail. Have you tried this approach successfully? Can you say more? $\endgroup$– AndrewDec 18, 2015 at 15:22