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I was tutoring a student on early trigonometry. Solving for the hypotenuse of a right triangle, but with sine, not Pythagoras.

The student went through getting the sine of a 45° triangle, and gave me the correct answer to 3 digits after decimal. (Note, the two legs of the triangle were both 7)

My issue – she had 0.7071 as sine of 45°, but lost the radical as she got the decimal solution. Her answer 7.071 was right, but I felt that $5\sqrt{2}$ was a missed opportunity. I accept and understand there‘s a time for calculators and 3 digits, but felt that early on, one should keep radicals, and both understand and be able to show the numbers for the sines of 0°, 30°, 45°, 60° and 90°.

The question - given that the US is going toward a common core approach, and given that students are allowed calculators early on, is there value to the pedagogical process to insist on keeping the irrational numbers through the solution? I am suggesting there will be greater understanding by keeping it, not indefinitely, but for understanding until the next level of math is introduced.

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    $\begingroup$ I am 51. If I recall, the 4 function calculator was still $100+ in 1977. I'm ok with calculators, but only at a certain point. The 1/1/sqr2 and 1/sqr3/2 triangles should first be learned, appreciated a bit, and then move on. $\endgroup$
    – JTP - Apologise to Monica
    Mar 14, 2014 at 16:18
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    $\begingroup$ This is not a question. Please note that the format on Stackexchange is that you have to ask specific questions that have answers. $\endgroup$
    – Thomas
    Mar 14, 2014 at 17:09
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    $\begingroup$ If you are unable to help with an edit, that makes the question nature clear, I'm good to close. I hoped for a "how to teach (the topic)" or "when to shift". If only because I've observed preferences from teachers, i.e. some saying "keep answer in radical form" and others "answers to 3 decimal places." $\endgroup$
    – JTP - Apologise to Monica
    Mar 14, 2014 at 20:14
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    $\begingroup$ "given that the country is going toward"?! Which country? $\endgroup$
    – quid ♦
    Mar 14, 2014 at 22:17
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    $\begingroup$ @Skytso - because her class didn't get there yet. As I wrote 'early trig,' this was the very first chapter, SOHCAHTOA, 45,45,90, and 30/60/90. radians were a couple chapters later. $\endgroup$
    – JTP - Apologise to Monica
    Mar 15, 2014 at 10:14

6 Answers 6

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My strongly held opinion is that some exact solutions are conceptually fundamental. Knowing that $\sin 45^\circ = \frac{1}{\sqrt{2}}$ (rather than approximately 0.7071) may not be important for applications in engineering — although it is certainly necessary in higher mathematics, and I wouldn't be surprised to see it in, say, theoretical physics — but it is an indicator of conceptual understanding. Even if you're only going to be an engineer and do numerical calculations, you need to understand the meanings of mathematical concepts in order to know what numerical calculations to do.

I would argue that if you don't know that $\sin 45^\circ = \frac{1}{\sqrt{2}}$, you don't really understand the meaning of the sine function and of the pythagorean theorem. The dividing line is a bit fuzzy — I'm not entirely sure on which side of it $\sin 30^\circ = \frac{1}{2}$ falls — but when learning trigonometry, I think $\sin 45^\circ = \frac{1}{\sqrt{2}}$ is definitely on the hither side. The position of the line also depends on the class; when teaching calculus I don't insist on $\sin 45^\circ = \frac{1}{\sqrt{2}}$, since it's not my job to be teaching them trigonometry any more, but I do still insist on some exact answers like $\sin \frac{\pi}{2} = 1$ and $\cos 0 = 1$ and $e^0=1$.

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    $\begingroup$ +1 - You had me at "My strongly held opinion is that some exact solutions are conceptually fundamental." I appreciate your answer. $\endgroup$
    – JTP - Apologise to Monica
    Mar 15, 2014 at 11:14
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    $\begingroup$ Someone could understand sine and cosine well enough to draw a picture of the unit circle, explain how sine and cosine are defined, sketch the graphs of sine and cosine, and explain the pythagorean identities, all without stopping to think that they might be able to compute $\sin(\frac{\pi}{4})$ or $\sin(\frac{\pi}{6})$ exactly. And if a student had such conceptual understanding, and had even computed these values before, they might not memorize the result. $\endgroup$
    – Steven Gubkin
    Mar 16, 2014 at 13:56
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    $\begingroup$ Stopping to think about how concepts relate to numbers is part of conceptual understanding. As is being able to recognize an expression whose value ought to be clear for conceptual reasons, even if you don't immediately remember its value off the top of your head. $\endgroup$
    – Mike Shulman
    Mar 16, 2014 at 15:12
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While exact solutions involving radicals are estetically pleasing to the mathematical mind, they are next to useless for practical use (Newton was thrilled to have found an easy way of approximating roots with the generalized binomial theorem!). Besides, few angles give such pleasing results.

It is a sad fact that what is required most of the time is just a precise enough numerical value.

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Some people would consider $1/\sqrt{2}$ as problematic or even wrong (in the sense of incomplete), as it should be further simplified to $\sqrt{2}/2$.

The reason why I consider this as related is that it is first of all a matter of convention.

It is possible to imagine a situation where the expected solution is the decimal approximation and the exact solution would be considered as 'wrong,' e.g., when the point would be practicing how to use a calculator.

In my mind there are certainly reasons to insist on students knowing or being able to derive the exact value of $\sin 45^{\circ}$, mainly as it requires an understanding of the trigonometric functions. Yet, if one wants to know if they know this, they should be asked explictly, or it should be explicitly clear that this is expected.

To me a main 'problem' in the situation seems to be to pose an exercise designed to use trigonometric functions where the angle is 45°. This seems a bit artificial, and it are often artificial exercises that lead to students' confusion.

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  • $\begingroup$ My original question had sides of 7,7,7√2 but it was changed as I messed up the latex, and the kind editor changed it to the simpler triangle. I will edit back a bit. $\endgroup$
    – JTP - Apologise to Monica
    Mar 14, 2014 at 22:57
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    $\begingroup$ Writing $\frac{1}{\sqrt{2}}$ as $\frac{\sqrt{2}}{2}$ isn't a simplification (although they are equal). $\endgroup$
    – petehern
    Mar 17, 2014 at 5:07
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    $\begingroup$ @petehern It is true simplification is a vague term, perhaps I should have put it under scare-quotes. However, my point stands, which is that some instructors would insist on such expressions involving radicals being given using rational (or integral) denominators. Depending on how one looks at the situation one can however consider this as a simplification. Every rational expression in $\sqrt{2}$ with rational coefficients can be written in a unique way as $a + b \sqrt{2}$ with rationals $a,b$ and one can consider this as a standard form of such an expression, and one 'simplifies' to it. $\endgroup$
    – quid ♦
    Mar 17, 2014 at 9:16
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It seems that it would be beneficial to know both the radical expression and the numerical answer. The radical expression is better for theoretical understanding, the numerical one for computational purposes and application.

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The introduction of calculators has turned calculus, algebra, and trigonometry into "which buttons do I push" and questions like "Why am I getting 0.8509 for the sine of 45 degrees?" It is, probably unintentinally, de-emphasizing the importance of theoretical mathematics.

The sine of 45 degrees is 0.7071 to four significant digits. This is not the same thing as saying that the sine of 45 degrees is 0.7071. There are times and places where that distinction is important. Students should understand that.

Addendum

Recently I was tutoring a student in geometry. She had been given the graph of a line (with two "nice" points indicated) and was asked to find the equation of the line. She came up with $y=2x+5$. I asked her to look at the line and tell me what the slope was. Using the two points, she came up, correctly, with $-2$. I then asked her what was wrong with her answer. It took more than a few heartbeats for her to figure out what was wrong. Would it be right to argue that she could have plugged the two coordinates into Wolfram Alpha and gotten the equation of the line, so why bother teaching her analytical skills? There is a mathematical difference between $1.414$ and $\sqrt 2$. It's the job of a mathematics student to understand that at more than a superficial level.

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  • $\begingroup$ I appreciate the feedback, although much of your answer seems in response to other's answers or comments. $\endgroup$
    – JTP - Apologise to Monica
    Nov 2, 2015 at 13:57
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    $\begingroup$ Some of those comments were mine! $\endgroup$
    – Steven Alexis Gregory
    Jan 28, 2017 at 15:47
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Whether you want a student to know $\sin(\pi/4)=1/\sqrt{2}$ depends on the goal of teaching trigonometry.

A century ago -- before radar, before aerial maps, before the wonders of modern calculation -- trigonometry was useful for surveying and navigation and more. That may explain its historical position in the high school curriculum. Nowadays, trigonometry does not deserve anywhere near a dedicated semester of high school math.

I would advocate teaching the sine function as the key example of a periodic function. If a high school student can answer the question "where does a circle centered at the origin intersect the line $y=x$?", it doesn't matter to me if they answer numerically or algebraically. But I'd rather that U.S. high schools stop with trigonometry around there.

So I would rather that high schools stay away from the pretty formula in the question. Time on that could be better spent on other topics, such as probability and statistics.

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    $\begingroup$ In spite of technological advances and aids to calculation, trigonometry is still used by surveyors, engineers, and cartographers, by physicists and in calculus (where integration by trigonometric substitution is a technique for solving problems that otherwise have very little to do with geometry). $\endgroup$
    – Confutus
    Mar 17, 2014 at 4:36
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    $\begingroup$ @Confutus: I agree, trigonometry is still used, and some college majors will require it. But I wouldn't teach it to most high school students. $\endgroup$
    – user173
    Mar 17, 2014 at 5:07
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    $\begingroup$ If the student uses trigonometry to answer the question you gave, I agree something went wrong in their education. $\endgroup$
    – quid ♦
    Mar 17, 2014 at 20:55
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    $\begingroup$ I hate hate hate hate hate the implication that theoretical mathematics is a waste of time for high school students and non science-oriented college students. As student fees become more important for a college's survival, student fees go up and admission standards go down. In my state, college students must pass an admissions test or or the equivalent of a high school algebra class in order to graduate. Some take that class three times before passing it. This creates a tremendous pressure to water down math education. Specifically, mathematical theory. $\endgroup$
    – Steven Alexis Gregory
    Nov 2, 2015 at 15:04
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    $\begingroup$ I don't think that every high school student should take a trigonometry class. But I think trigonometry is perfectly suitable for college-bound high school students. I also think that linear algebra should make its appearance in high school. Maybe even as an alternative to calculus. It would certainly be more "practical" to most students than calculus. $\endgroup$
    – Steven Alexis Gregory
    Nov 2, 2015 at 15:06

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