Limits of trigonometric functions

In summary, the conversation discusses the issue of certain trigonometric equations returning incorrect answers when entered into a TI-89 calculator. It is noted that the calculator should be in radian mode for accurate results, and that the units for angle measures can affect the output of the calculator. The conversation also touches on the importance of understanding and tracking units to prevent errors.
  • #1
OrbitalPower
Why do some problems return the wrong answer while others do not on the ti-89.

For example:

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

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

But

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

does not?
 
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  • #2
Not sure, but you should be able to do these easy by hand.
 
  • #3
Shows up wrong (ships up as pi over something).

What is the something? 3.14159...?
 
  • #4
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?
 
  • #5
I am very confused :( These problems have nothing to do with angles! It shouldn't matter what mode you shove these into your calculator.
 
  • #6
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.
 
  • #7
OrbitalPower said:
[tex]\[ \lim_{x \to 0} \frac{\cos\theta \tan\theta}{\theta}\][/tex]

does not?

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

[tex]\[ \lim_{\theta \to 0} \frac{\cos\theta \tan\theta}{\theta}\][/tex]
 
  • #8
OrbitalPower said:
Why do some problems return the wrong answer while others do not on the ti-89.

For example:

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

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

But

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

does not?

Not wrong

[tex]\[ \lim_{x \to 0} \frac{\cos\theta \tan\theta}{\theta}= \lim_{x \to 0} \frac{\sin\theta }{\theta}=\frac{\pi}{2 \arcsin 1}\][/tex]
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
 
  • #9
Make sure your calculator is in "radian" mode rather than "degree" mode!
 
  • #10
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?)
 
  • #11
OrbitalPower said:
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
 

Related to Limits of trigonometric functions

1. What are the basic trigonometric functions?

The basic trigonometric functions are sine, cosine, and tangent. These functions represent the ratios of the sides of a right triangle, and are commonly denoted as sin(x), cos(x), and tan(x), respectively.

2. What is the domain of trigonometric functions?

The domain of trigonometric functions is all real numbers, as the functions can take any angle as an input. However, the range may be limited depending on the specific function and the range of values for the angle.

3. What are the limits of trigonometric functions?

The limits of trigonometric functions depend on the specific function and the value of the angle. For example, the limit of sin(x) as x approaches 0 is 0, while the limit of tan(x) as x approaches pi/2 is undefined.

4. How do I find the limit of a trigonometric function?

To find the limit of a trigonometric function, you can either use calculus techniques such as L'Hopital's Rule or evaluate the function at values approaching the desired limit. Additionally, you can use the fundamental trigonometric identities to simplify the function and find the limit.

5. What is the significance of the limits of trigonometric functions?

The limits of trigonometric functions are important in understanding the behavior of these functions as the input approaches certain values. They can also be used to solve problems in calculus and other areas of mathematics, such as finding the maximum or minimum value of a function.

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