# A student's problem on solving simple trigonometric operations

In my class today as I was checking homework of trigonometric operations, I called one of the students and gave her one of the problems of book which was $\frac{1}{(1-\cot (\theta ))}+\frac{1}{(\cot (\theta )-1)}$. The question was to simplify that problem so I told her that the question is in bit complex form so then she should make it simpler. In the process I told to take common (-) from the 2nd term of $\cot(\theta)-1$ so then it becomes simple to take LCM and solve the problem. As it seems she seems to have problem on algebra. So it seems like I have to start from the beginning but I do not know from where to start. Are there any systematic ways to make her understand all those things easily? A specific way to make understand and able her like student to solve those kind of problems?

• If my edit is not correct, please reverse. This appear to be a pretty simple observation that a-b and b-a are opposites, and add to zero. No trig observation really needed. – JoeTaxpayer May 10 '14 at 16:00

Yes. Repeatedly break the problem down by removing one layer of complexity at a time until you discover a problem she can complete.

The basic layer of this problem is fractions. Ask her if she can add these fractions:

$\frac13 + \frac17$.

The next layer of this problem is fractions within fractions. Ask if she can simplify the following expression:

$\frac1{\frac13 + 1}$

The next layer of this problem is rewriting trigonometric functions. For example, can she complete this problem?

Simplify $\tan x \cdot \cos x$.

You will probably find that she is able to complete at most one, if any, of these subproblems. Asking these simpler problems is very productive. Ideally you will find one or two of these skills that she does not have. Then you can get out a blank sheet of paper, and together, cover it with examples of that one skill.

By the time you've touched all the skills, she should be able to complete that problem. And she should leave with a sheet of paper for each skill she practiced that day. Even if it starts with adding fractions.

• Chris - I may be stating the obvious, but don't we have a far deeper problem if a trig student hasn't yet mastered the first two examples above? How is a teacher supposed to teach a semester of material if this much effort to review grade school math is required? – JoeTaxpayer May 9 '14 at 11:47
• @JoeTaxpayer An interesting discussion question in its own right, I think. My current (non-authoritative) understanding of "basic skills" is that students in general do not master skills when they first learn them. In your words, we almost always have "far deeper problems." :) Students master skills when they are successfully used in bigger problems like these. Acknowledging that many of my calculus students don't quite get logarithms, trigonometry, or even fractions is often the most useful first place to start! – Chris Cunningham May 9 '14 at 16:30
• I don't think I made this clear in my comment: I do not think a trig student who can't immediately handle $\frac{1}{\frac13 + 1}$ is doomed, and I don't even see it as a particularly sad state of affairs. We are definitely in trouble if it takes very long for them to pick it up again, but complete mastery? Probably not necessary yet. – Chris Cunningham May 9 '14 at 16:36
• Doomed is certainly an overstatement, I suppose it's a matter of degree. Is the fraction refresh 5 minutes, or a week's worth of classes? Your 3rd example is on the list of questions I see under the "10 questions in 5 minutes" drill page before the longer review. It should be at a glance, not a recap every time. (But I do respect the need to have some recap along the way) – JoeTaxpayer May 9 '14 at 16:55

I tend to channel Polya in such situation and try to guide by asking questions like: "How do you add two fractions?", "Are the denominators completely different or can we find similarites (which we could exploit)?"

Colors usually help in discovering the structure.

$$\frac{1}{(\color{green}1\color{black}-\color{red}\cot (\theta )\color{black})}+\frac{1}{(\color{red}\cot (\theta )\color{black}-\color{green}1\color{black})}$$

"How can we get from 'green minus red' to 'red minus green'?"

It's similar to what Chris Cunningham suggested, but I try to stay with the given problem as long as possible. Task switching is easy for the teacher, since he knows that "all this is the same", but often a nontrivial job for a struggling student.