Trig questions



#2

I was reviewing the Coffin Algebra and Pre-calculus book for the 48GX and saw some trig identities and wondered about some of them. The first thing I see is some of the identities have = signs with three lines instead of two. Anyone know the significance of that?
Also, the half angle identities, when you take the sine of say, 20 degrees you get a number, a single number, and if you take the sine of half of that, sine 10 degrees, you get another single number. But when you look at the half angle identities, it goes +/- square root, etc, so you end up with two solutions. How can that be an identity when the original operation has only one solution?
Sin A/2 =( and for this one the = sign is a regular 2 bar one, why?)
+/- Sqrt (1-cos A/2)

So how can the dual solution value be the same as just dividing the angle by two and taking the sine which only has one solution?

Also as an aside, does anyone use the Mathpro card by Da Vinci Technologies group? How would you grade it compared to other math cards? Are there better ones, if so, which ones?
I have a Mathpro card installed in my 48GX and it sure seems powerful to me but don't know the extent of the competition. Thanks all. Don.


#3

Quote:
The first thing I see is some of the identities have = signs with three lines instead of two. Anyone know the significance of that?

Don,

In mathematics, the 3 horizontal bars mark identities, while the common = sign is for equations.

HTH,

Walter


#4

So I guess that means the regular = sign were typo's because they were in the identity section.


#5

Don,

AFAIK something can't be identical to 2 *different* things (+/- !).

#6

This is not a comment to Walter, but the Forum s/w will let me respond to this post (see "HTML glitch" above):

(BTW, I appended "#143984" to the URL to make this work)


Quote:
But when you look at the half angle identities, it goes square root, etc, so you end up with two solutions. How can that be an identity when the original operation has only one solution?

Sin A/2 = Sqrt (1-cos A/2)

So how can the dual solution value be the same as just dividing the angle by two and taking the sine which only has one solution?


Don --

The difference between triple-bar identity sign and the double-bar equation sign has been explained. Really, they are all identities, and should be consistently denoted.

The equation you listed:

Sin A/2 = Sqrt (1-cos A/2)

is not really a half-angle formula, as both angles are the same. Make it "cos2 (A/2)" and it's correct, but only as a form of

sin2 x + cos2 x = 1

The equation should have been listed as

Sin A/2 = Sqrt ((1-cos A)/2)

and your reference should have stated the proper sign of the square root, based on sector of the angle.

BTW, an angle "phi" resulting from an inverse-trigonometric ("arc") function will range between -180o > phi >= +180o

-- KS


Edited: 23 Nov 2008, 3:42 p.m. after one or more responses were posted


#7

I think I understand now, the mistake was in the book, in not more clearly specifying the quadrant. That makes more sense. It was driving me a bit batty when I knew there was only one sign for one sine as it were:)

#8

The +/- sign depends on what quadrant A/2 is in (or where A itself is); eg., if A is in (0,pi) both sine and cosine of A/2 are positive; if A is in quadrant III, cosine A/2 is negative and sine A/2 is positive, etc.


#9

Quote:
The +/- sign depends on what quadrant A/2 is in (or where A itself is); eg., if A is in (0,pi) both sine and cosine of A/2 are positive; if A is in quadrant III, cosine A/2 is negative and sine A/2 is positive, etc.

Yes but a single angle will have a single sign, 1 to 179 deg= +
and 181 to 359 = -
So why would there be a +/- sign in the equation, if you input 10 degrees you should always and only get a + sign it seems to me.

#10

I suppose I regard the "formula" as a reminder for calculating the sine or cosine of half an angle, rather than it telling me there are two "results" using the formula (like the quadratic formula).

One solution would be to not do the last step of the derivation for the formula and leave it "cos(A/2)^2=(1+cos(A))/2", or add a signum factor to the formula:

cos(A/2) = (-1)^(int[(A+pi)/(2pi)] sqrt[(1+cos A)/2]

but that seems a little extreme.

#11

Test for KS. Thanks.


Possibly Related Threads...
Thread Author Replies Views Last Post
  Trig vs hyperbolic handling differences in Prime CAS Michael de Estrada 3 215 11-08-2013, 06:26 PM
Last Post: Mark Hardman
  Trig Functions Howard Owen 11 513 09-16-2013, 02:53 PM
Last Post: Fred Lusk
  trig scales on the Post Versalog slide rule Al 12 518 09-15-2013, 06:01 AM
Last Post: John I.
  Some trig help, please Matt Agajanian 33 967 04-12-2013, 06:39 PM
Last Post: Matt Agajanian
  Trig Identity question Namir 13 492 03-04-2013, 07:45 PM
Last Post: Eric Smith
  Question about trig functions approximation Namir 8 350 01-10-2013, 06:49 AM
Last Post: Valentin Albillo
  WP 34S Complex trig/hyperbolic bugs? Eamonn 35 1,027 04-26-2012, 07:32 PM
Last Post: Pete Wilson
  Just a thought I had--a 35S Trig fix Matt Agajanian 62 1,135 03-26-2012, 10:40 AM
Last Post: bill platt
  10bii+ Trig Bug Katie Wasserman 5 239 07-06-2011, 05:01 PM
Last Post: Thomas Radtke
  HTML glitch in Don Jennings' "Trig questions" thread? Karl Schneider 5 190 11-22-2008, 04:14 PM
Last Post: V-PN

Forum Jump: