CAS capability question

[JayG]

Is CAS not designed to simplify trig functions with more than one
operand or am I doing something wrong.

As expected, CAS returns true for (sin(x))^2=1-(cos(x))^2. So I
expected the same conclusion using multi-operand functions, but other
identities like cos(2*x)=(cos(x))^2-((sin(x))^2, do not return true.

What am I doing wrong?

Or if the calculator is not meant to resolve equations like that, what
rules define the limits of CAS?

[Paul Alves]

Try this: texpand(cos(2x))=cos(x)^2-sin(x)^2 and you will get the true
result.

This is the trig expand command. Also accessible through the menus:
Menu>Algebra>Trig>Expand.

Paul

[Joe]

You are not doing anything wrong. The TI cas recognizes the
fundamental identities and variations thereof as being true but has
never recognized the double angle formulas as being true.

[Jay Gourley]

Okay, the how about this? CAS says
sin(2*pi/17)=2*sin(pi/17)*cos(pi/17) is false.

It's not just failing to recognize an identity. It's reporting the
identity is a contradiction.

Please tell me I'm doing something wrong.

CAS may use incorrect significance since it calculates a value for the
left side thad is different from the right side by 1*10^-14. Do you
suppose that's the reason?

BTW, you can substitute other digits for the "17" and get the same
false result.

[Joe]

"It's not just failing to recognize an identity. It's reporting the

identity is a contradiction." I tried substituting a few values for
the 17, and for some it returns true and for others it returns false.
For example with sin(2*x)=2*sin(x)*cos(x) on my v200 which should be
the same cas as on the nspire cas, the original equation is returned.
Then substituting Pi/5 for x returns false but substituting Pi/4
returns true. Have you brought it to TI's attention? If so, what
sort of response did you get?

[Nelson Sousa]

The algorithms to expand/collect trig functions are not very powerful.
It could be checking whether the values are numerically equal and the
trig functions have a precision of around 10^-12, so the results could
indeed be different.

Nelson

[JayG]

The numerical results accuracy is reasonable, the calculator should
know it's own limit. CAS reports some identities as contradictions,
possibly based on false discrepancies beyond the accuracy.

I'll call TI today and let you know what it says.

[Wayne]

Jay,
As you and Nelson say, your result is probably attributable to
numerical accuracy of the system. More generally though, the result
occurs because the CAS evaluates sin(2*pi/17) before it does the
expansion of the trigonometric identity. Experienced programmers of
the CAS have known for some time that the order of substitutions and
evaluations make a big difference in results. In order to obtain the
correct result from the CAS, you must force the CAS to make the
substitutions and expansions in the "right" order. Here is the way I
would have done that:
expr(string(expr("texpand(sin(2x))"))&"|x=π/17")=2*sin(((π)/(17)))*cos
(((π)/(17))) which the CAS evaluates to "true". I won't attempt to
explain what is going on here, expecting that you and others can parse
the construction by using the reference guide. Suffice it to say that
CAS programmers who regularly use this kind of construction would do
something like the above to obtain the correct result. It is not
ideal, but looking for prefection in a CAS will give you a lot of grey
hair. Mine is already all grey. :-)
Wayne

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