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### Rendering output of trigonometric functions as fractions ### Rendering output of trigonometric functions as fractions

by Sergio Chaves -
Number of replies: 2

I am writing a question where I want to display the values $$\sin(\theta)$$ and $$\cos(\theta)$$ as fractions, so students are able to identify the representative angle that they are coming from.

I am using the Fractions Context, but I do not succeed after evaluating the functions. A sample code can be found below. As you notice, I tried  several formats but I get displayed either "0.5" or " $$\sin(\pi/6)$$" for example. Is there a way to achieve this without using arrays to have the fractions stored? I want to avoid this approach as I would like to randomize the angle to either $$\pm k \frac{\pi}{6}$$ or $$\pm k \frac{\pi}{4}$$.

DOCUMENT();
loadMacros(  "PGstandard.pl",  "MathObjects.pl",  "PGcourse.pl",  "PGML.pl",  "parserPopUp.pl","PGchoicemacros.pl","contextFraction.pl",);TEXT(beginproblem());$showPartialCorrectAnswers = 1;Context("Fraction-NoDecimals");$theta = list_random(Formula("0"),Formula("pi/6"), Formula("pi/4"),  Formula("pi/3"), Formula("pi/2"));$a11 = Compute(cos($theta));$a12 = -sin($theta);$a21 = Formula(sin($theta));$a22 = Formula(cos($theta));Context()->texStrings;BEGIN_PGMLThe following matrix is the rotation matrix with angle [\theta = [$theta] ].[ \begin{bmatrix}[$a11] & [$a12] \\[$a21] & [$a22]\end{bmatrix}]END_PGMLENDDOCUMENT(); In reply to Sergio Chaves ### Re: Rendering output of trigonometric functions as fractions by Alex Jordan - I have an algorithmic routine for handling this. It's half baked into a macro library and I just need time to clean it and test it before contributing it. If you *know* you have a number that is output from sine or cosine applied to a nice angle (multiple of pi,pi/2,pi/3,pi/4,pi/6) then you know that squaring that number gives you a rational number. For example, if$y = cos(pi/6), then $y**2 is 3/4. It might be a floating point real that is only close to 3/4, but that is OK with what comes next. The Fraction context will turn$y**2 into a clean Fraction. So $y2 = Fraction($y**2) is now the Fraction object 3/4.

And now
($n,$d)=$y2->value; makes$n the numerator and $d the denominator. If$n==0, return Real(0).

Else if $d==1, that implies$n must be 1. Now return Real(1) (see *).

Else $d must be 4, and$n is among 1,2,3. If $n==1, return Fraction(1,2) (see *). Else$n==2 or $n==3. Return Formula("sqrt($n)/2"), with context flag reduceConstantFunctions=>0 (see *).

*Always modify the output according to the sign of the original \$y.

Now this much could just as easily be put into a hash that you keep in a macro library, instead of being algorithmic. But what I really have handles general rational multiples of nice trig values as well. It's just a little more complicated. 