Help With Identities

[PeaceMaker1050]

Hello All,

I have recently come across a math problem while doing homework that I don't understand. The problem is "tan30=2tan15/1-tan^2(15). Find the exact value for tan15, giving your answer in the form a-(sqrt)b, where a and b are positive integers." I would appreciate any help with this problem.

 

[Deleted member 4993]

Hello All,

I have recently come across a math problem while doing homework that I don't understand. The problem is "tan30=2tan15/1-tan^2(15). Find the exact value for tan15, giving your answer in the form a-(sqrt)b, where a and b are positive integers." I would appreciate any help with this problem.

Hint:

tan(30^o) = 1/sqrt(3)

 

[pka]

The problem is "tan30=2tan15/1-tan^2(15). Find the exact value for tan15, giving your answer in the form a-(sqrt)b, where a and b are positive integers." I would appreciate any help with this problem.

I have sworn off dealing anyone so out-of-date as to use degrees. But this is an interesting question,
$\displaystyle \tan\left(\dfrac{\pi}{6}\right)=\dfrac{\sqrt{3}}{2}$ now let $\displaystyle y=\tan\left(\dfrac{\pi}{12}\right)$
Therefore the problem becomes $\displaystyle \dfrac{\sqrt{3}}{2}=\dfrac{2y}{1-y^2}$
Can you solve for $\displaystyle y~?$

 

[lookagain]

The problem is "tan30=2tan15/1-tan^2(15). Find the exact value for tan15, giving your answer in the form a-(sqrt)b, where a and b are positive integers."

You are missing degrees and grouping symbols.

tan(30^o) = 2tan(15^o)/[1 - tan^2(15^o)]

$\displaystyle \tan\left(\dfrac{\pi}{6}\right)=\dfrac{\sqrt{3}}{2}$
Therefore the problem becomes $\displaystyle \dfrac{\sqrt{3}}{2}=\dfrac{2y}{1-y^2}$

No, it doesn't. $\displaystyle \ \ tan\bigg(\dfrac{\pi}{6}\bigg) \ = \ \dfrac{\sqrt{3}}{3}$.

I have sworn off dealing anyone so out-of-date as to use degrees.

No, you clearly did not "swear off dealing anyone so out-of-date as to use degrees."
You replied to that question.

It's not out-of-date. Using degrees or radians is a preference. What you stated is irrational.
What you have stated about it in the past is irrational. If you continue to do so, you would remain irrational about it.

 

[topsquark]

Math tends to use radians and non-Calculus Physics tends to use degrees.

-Dan

 

[Deleted member 4993]

I did not want to get into this ....

But since most of the commercial protractors are marked with degrees - engineers (being the practical persons they are) tend to use degrees....

 

[Otis]

Math tends to use radians …

In the U.S., it seems that math courses introducing concepts of angles and their measurement tend to teach degree measure first. Students learn about radian measure later, after which exercises are assigned using degrees or using radians. Later still, some students learn radian methods that don't work using degrees. Such is life (in the U.S., anyway).

 

[Steven G]

Math tends to use radians and non-Calculus Physics tends to use degrees.

-Dan

non-Calculus Physics. Calculus was invested to do Physics! Do I need to say more.

 

[topsquark]

non-Calculus Physics. Calculus was invested to do Physics! Do I need to say more.

I agree to a point. Yes, when Calculus came about it made more sense to talk about angles in radians. (I don't know when the unit "degree" came about.) But not all people taking Intro Physics is going to move on to the higher level courses. (Nurses, for example.) Since, at least in the US teaching system, degrees is the usual angular unit they are taught in secondary school they do Physics in degrees. There are some exceptions to the "rule": when dealing with angular speeds, for example, rad/s is usually used. But most topics such as Kinematics use degrees.

Thpppptt!

-Dan

 

[pka]

non-Calculus Physics. Calculus was invested to do Physics! Do I need to say more.

Jomo, I like your comment. Some forty-five years ago I was a new faculty member in charge of the mathematics service courses at a regional university in the American south. Well I chose to use my all time favorite calculus textbook: Calculus by Gillman&McDowell. To cut to the chase, when the chair of Chemistry&Physics realized that the book taught that the $\displaystyle \log(x)$ meant the natural logarithm, well you know what hit the fan. Lucky for me the provost at the time was a mathematician. But the worm turns. Some fifteen years on, I was chair of Mathematical Sciences and the same chair came to me for help with her new calculator. It seems it was giving her incorrect results. I pointed out that log function returned natural values and the trig functions assumed radian values. She said to me, "didn't we have this debate some ten years ago?"

 

[JeffM]

I agree to a point. Yes, when Calculus came about it made more sense to talk about angles in radians. (I don't know when the unit "degree" came about.) But not all people taking Intro Physics is going to move on to the higher level courses.

Indeed, some people never take physics or calculus at all. But their sort are from the lower orders, quite out of date, not at all of the quality that a mathematician should be expected to deal with.

Basing angular measurement on 360 may go back to Hipparchus, who probably found working with a number divisible by 2, 3, and 5 more computationally convenient than working with Archimedes' limits on the value of pi. And certainly Hipparchus could not be more out of date: he has been dead for millennia.

 

[Steven G]

Jomo, I like your comment. Some forty-five years ago I was a new faculty member in charge of the mathematics service courses at a regional university in the American south. Well I chose to use my all time favorite calculus textbook: Calculus by Gillman&McDowell. To cut to the chase, when the chair of Chemistry&Physics realized that the book taught that the $\displaystyle \log(x)$ meant the natural logarithm, well you know what hit the fan. Lucky for me the provost at the time was a mathematician. But the worm turns. Some fifteen years on, I was chair of Mathematical Sciences and the same chair came to me for help with her new calculator. It seems it was giving her incorrect results. I pointed out that log function returned natural values and the trig functions assumed radian values. She said to me, "didn't we have this debate some ten years ago?"

Great story!

 

[Deleted member 4993]

Indeed, some people never take physics or calculus at all. But their sort are from the lower orders, quite out of date, not at all of the quality that a mathematician should be expected to deal with.

Basing angular measurement on 360 may go back to Hipparchus, who probably found working with a number divisible by 2, 3, and 5 more computationally convenient than working with Archimedes' limits on the value of pi. And certainly Hipparchus could not be more out of date: he has been dead for millennia.

I thought Sumerians first introduce division of a circle into 360 parts then to Egyptians then to those "irrational" Greeks.

But my knowledge in this is strictly hearsay - I do not know this to any depth....

 

[JeffM]

I thought Sumerians first introduce division of a circle into 360 parts then to Egyptians then to those "irrational" Greeks.

But my knowledge in this is strictly hearsay - I do not know this to any depth....

Subhotosh

It is certainly true that the Sumerians had a numeration system based on 60 and did astronomy. It would therefore not be at all surprising had they created a standard that divided the circle into 360 segments.

The problem with such historical research is two fold: (1) what exactly are we asking, and (2) what evidence remains. How much of their mathematics the Greeks learned from the Phoenicians is unknown. The Greeks certainly did not inherit anything directly from the Sumerians, whose language was an isolate and who disappeared from the historical record 2000 years before the Greeks are recorded doing much mathematics. But the Semitic civilizations of the ancient Near East were direct inheritors (or perhaps robbers) of the Sumerians, and the Greeks must have been in contact with the Semites, especially the Phoenicians, from when they first reached the Aegean Sea. Even an attribution to Hipparchus is uncertain because none of his work has survived, merely references to it.