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 ▼ Don Jennings Junior Member Posts: 46 Threads: 6 Joined: Oct 2008 11-20-2008, 05:42 PM 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. ▼ Walter B Posting Freak Posts: 4,587 Threads: 105 Joined: Jul 2005 11-20-2008, 06:49 PM 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 ▼ Don Jennings Junior Member Posts: 46 Threads: 6 Joined: Oct 2008 11-22-2008, 09:15 AM So I guess that means the regular = sign were typo's because they were in the identity section. ▼ Walter B Posting Freak Posts: 4,587 Threads: 105 Joined: Jul 2005 11-24-2008, 01:20 AM Don, AFAIK something can't be identical to 2 *different* things (+/- !). Karl Schneider Posting Freak Posts: 1,792 Threads: 62 Joined: Jan 2005 11-22-2008, 03:02 PM 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 ▼ Don Jennings Junior Member Posts: 46 Threads: 6 Joined: Oct 2008 11-23-2008, 01:26 AM 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:) Chuck Senior Member Posts: 320 Threads: 59 Joined: Dec 2006 11-20-2008, 10:50 PM 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. ▼ Don Jennings Junior Member Posts: 46 Threads: 6 Joined: Oct 2008 11-21-2008, 06:11 PM 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. ▼ Chuck Senior Member Posts: 320 Threads: 59 Joined: Dec 2006 11-21-2008, 08:02 PM 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. ▼ Pal G. Senior Member Posts: 260 Threads: 21 Joined: Nov 2006 11-22-2008, 08:31 AM Test for KS. Thanks.

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