Calculator Techniques for Trigonometry

Finding values on your calculator is for the most part pretty straight forward. You just key in the number and push the button, but there are a few things to look out for.

cot, sec, csc

Problem: They aren't there.
Solution: You have to use their reciprocals and then the 1/x button.
 

function    keys

cot	tan, 1/x
sec	cos, 1/x
csc	sin, 1/x

Very Large Values

Problem: Some calculators don't like them. And here I am not talking about the kinds of numbers you need scientific notation in order to put into the calculator, I'm talking about such things as 1000 radians even. Since 2pi is only around 6, this represents a lot of rotations around the circle. I'm not sure why calculators have troubles with such things. It may have something to do with the power series that are used by them to compute trig functions. The larger the value you use, the more terms of these series are needed to get an accurate enough answer, and then with lots of terms it needs to compute large powers of the large values, and this could overflow it.

Solution: Subtract off a large enough multiple of 2pi or 360° so that the calculator is happy.

Inverses

To find a value whose sin, cos, or tan is a give number is not difficult. You use the inverse trig function for this. There are various notations for this on different calculators. You may have an INV button or you might have buttons with the trig function and a -1 exponent to indicate this. Remember this doesn't mean the reciprocal, it means the inverse function. Another possibility is that you will have arcsin, arccos, and arctan. The idea behind that notation is 'the arc whose ____ is'. In any case at least for sin, cos, or tan, all you have to do is enter your number and push the button.

Inverse cot, sec, csc

Problem: Again they aren't there.

Solution: Again you have to use their reciprocals, but here it is slightly more complicated.

Suppose you are trying to find theta in the first quadrant so that sec(theta)=3. What you need to do is figure out what this tells you about cos(theta) and work from there.

This means that you need to first push the 1/x key and then the reciprocal function.

The Tricky Problem, Putting Theta in a Given Interval

You may have noticed that at the beginning of this section I wrote the word 'a' in boldface and that up until now I have been only talking about finding values in the first quadrant. Well, there is a reason for that. The problem with talking about finding the value whose sin, cos, or whatever is a given number is that there are lots of them. So to narrow things down you are usually told which quadrant to look. This is easy provided the value you want is in the same place as your calculator is looking. But what if it isn't? Before we can talk about that we have to find out where the calculator does look. The calculator uses the standard convention that is used in mathematics when defining the inverse trig functions, which is this. This means that if you are given a positive number and you want to find a value in the first quadrant, the calculator will give you what you want with no problem. But otherwise, you have to be a bit careful. If if were just the matter that the functions are periodic, then it would be easier, because all you would have to do is add an appropriate multiple of 2pi. But even within one go around the circle, for most values there are 2 places where each of these functions is equal to it. So what you have to do is remember the identities that give you these. For tangent this is easiest. It is periodic with period pi, so all you have to do is find the right multiple of pi to add. But for sine and cosine you have to remember the identities that tell you that the sine of the supplementary angle is the same as the sine of the original and that the cosine of the minus is the same as the cosine of the original. You may find it helpful to keep in mind this picture when trying to remember these identities. Pi-theta is the reflection across the y axis, so since the sine is the y coordinate, it is the same there. -Theta is the reflection across the x axis, so since cosine is the x coordinate, it is the same there. Theta+ or -pi is the reflection across the original. That makes both the sine and the cosine opposite there, so it makes the tangent the same. (See also How to Test a Relation for Symmetries.)

All together this means that to get the value where you want it you are allowed the following operations, where theta denotes the original value and n is any integer.

The thing to do is first adjust to get it into the right quadrant and then add the right multiple of 2pi so that it has the right numerical value.

This will probably be clearer with an example, so if you have one that you are stuck on, please let me know. Perhaps I will add it to the article.

Shelley's Math Articles