Limits of trigonometric functions

[OrbitalPower]

Why do some problems return the wrong answer while others do not on the ti-89.

For example:

\[ \lim_{x \to 0} \frac{\cos\theta \tan\theta}{\theta}\]

Shows up wrong (shows up as pi over 180).

But

\[ \lim_{x \to 0} \frac{\sin x(1 - \cos x)}{2x^2}\]

does not?

 

[rocomath]

Not sure, but you should be able to do these easy by hand.

 

[mathman]

Shows up wrong (ships up as pi over something).

What is the something? 3.14159...?

 

[OrbitalPower]

No. Pi/180. But that isn't one. My question is, why do certain trigonometric equations show up as the textbook answers, but not others.

For example, like I said, the second one returns correctly, the first one does not. I understand it should be in radian mode now, but why does degree mode give the right answer 50-75% of the time in my experience?

 

[Gib Z]

I am very confused :( These problems have nothing to do with angles! It shouldn't matter what mode you shove these into your calculator.

 

[OrbitalPower]

Right, Gib Z. That's exactly what I was thinking. Just thought it would be helpful for the forum if someone had a quick explanation.

 

[kentm]

\[ \lim_{x \to 0} \frac{\cos\theta \tan\theta}{\theta}\]

does not?

Nothing happens to that limit as x changes, maybe that's why your calculator comes up with something different.

\[ \lim_{\theta \to 0} \frac{\cos\theta \tan\theta}{\theta}\]

 

[lurflurf]

Why do some problems return the wrong answer while others do not on the ti-89.

For example:

\[ \lim_{x \to 0} \frac{\cos\theta \tan\theta}{\theta}\]

Shows up wrong (shows up as pi over 180).

But

\[ \lim_{x \to 0} \frac{\sin x(1 - \cos x)}{2x^2}\]

does not?

Not wrong

\[ \lim_{x \to 0} \frac{\cos\theta \tan\theta}{\theta}= \lim_{x \to 0} \frac{\sin\theta }{\theta}=\frac{\pi}{2 \arcsin 1}\]
in degrees pi/180 deg^-1
in grad pi/200 grad^-1
in rad 1 rad^-1
in mil pi/3200 mil^-1
in clock pi/6 hours^-1
in rotations pi/.5 rot^-1

Angle measure units matter
Rad make calculus things look nice
Why use the calculator at all save that for later

 

[HallsofIvy]

Make sure your calculator is in "radian" mode rather than "degree" mode!

 

[OrbitalPower]

Yeah, thanks guys. And I think I see what you're saying lurflurf. If you convert it from degrees to randians its 1 anyway. (What's the deg^-1?)

 

[lurflurf]

Yeah, thanks guys. And I think I see what you're saying lurflurf. If you convert it from degrees to randians its 1 anyway. (What's the deg^-1?)

It is from unit analysis

(10 feet)/(5 seconds)=2 feet seconds^-1

if
units(x)=degrees
units(sin(x))=1 (ie no units)
then
units(sin(x)/x)=1/degrees=deg^-1
angle measure units are not entirely well defined
but tracking them can prevent errors especially when radians are not being used

if anyone like -1 better than 2
pi/arccos(-1)=pi/(2 arcsin(1))
=limit x->0 sin(x)/x
for that matter may expressions are possible