compound angle problem

[Locus]

Good morning/evening everyone.

I'm going through a book riddled with mistakes and that causes me to doubt my math skills first, then the book and then the whole discipline of mathematics, and im dad, so you can imagine what it does to leaving cert teenage daughter.
This particular exercise already cost me almost a full day, so please help.
Exercise goes as follows:
Prove that:
cos(45degrees + theta) -(cos45degrees - theta) = -squarerootof2sintheta

I tried the same thing a few times and i consulted various sources on and offline so finally I came to the decision that they have made a mistake, and the exercise should have, instead, been:

Prove that:
cos(45degrees + theta) - cos(45degrees - theta) = -squarerootof2sintheta

Am I wrong in making that assumption and is there a way where the original exercise is right?

 

[Dr.Peterson]

Exercise goes as follows:
Prove that:
$$\cos(45^\circ + \theta) -(\cos45^\circ - \theta) = -\sqrt{2}\sin\theta$$
I tried the same thing a few times and i consulted various sources on and offline so finally I came to the decision that they have made a mistake, and the exercise should have, instead, been:

Prove that:
$$\cos(45^\circ + \theta) -\cos(45^\circ - \theta) = -\sqrt{2}\sin\theta$$
Am I wrong in making that assumption and is there a way where the original exercise is right?

I've made both copies of the equation readable above; at first I didn't even see the difference. But you are right; the first is nonsense, while the second is true.

 

[The Highlander]

I've made both copies of the equation readable above; at first I didn't even see the difference. But you are right; the first is nonsense, while the second is true.

Well spotted! I couldn't see the difference either, until your post, lol.

Sounds like the "dad" ought to get hold of a different "book".

 

[Locus]

I've made both copies of the equation readable above; at first I didn't even see the difference. But you are right; the first is nonsense, while the second is true.

Thank you Dr.Peterson for making it readable and for the answer.

 

[Locus]

Well spotted! I couldn't see the difference either, until your post, lol.

Sounds like the "dad" ought to get hold of a different "book".

It is the official book everyone in the country goes by, and it is full of mistakes. There is roughly one in every subchapter, either in the exercises or the actual theory. And I am not trying to smear anyone's name, but i am compiling a list and i am going to send them an email with it, and if they dont fix it, then i will go public.

 

[The Highlander]

It is the official book everyone in the country goes by, and it is full of mistakes. There is roughly one in every subchapter, either in the exercises or the actual theory. And I am not trying to smear anyone's name, but i am compiling a list and i am going to send them an email with it, and if they dont fix it, then i will go public.

What's the book [Title & Author(s)]?
What topics does it cover and to what depth (& at what level)?
Maybe a different textbook would be advantageous.
I could only offer a text in English which may be of little use if you're not in a English speaking country but your English does seem good so, if I have anything that meets the above criteria, I can pass it on to you.

 

[Locus]

Higher level leaving certificate maths. Im leaving out the authors names at the moment ?

 

[Locus]

What's the book [Title & Author(s)]?
What topics does it cover and to what depth (& at what level)?
Maybe a different textbook would be advantageous.
I could only offer a text in English which may be of little use if you're not in a English speaking country but your English does seem good so, if I have anything that meets the above criteria, I can pass it on to you.

And thank you for the offer. Just the name and author will do. Not for this specific task at hand maybe, but I do intend to keep my maths unrusted for the next generation hopefully.

 

[The Highlander]

Higher level leaving certificate maths. Im leaving out the authors names at the moment ?

And thank you for the offer. Just the name and author will do. Not for this specific task at hand maybe, but I do intend to keep my maths unrusted for the next generation hopefully.

I’m not entirely clear on what you mean in your responses. I was asking for the details I requested to judge whether I had any textbooks that might cover corresponding topics to the same or similar depth.

I had hoped to look at the content of the textbook you are using to determine this but, from what you have said, I suspect the best I can offer you is the two main texts we use to teach “Higher Maths” here in my country.

Just let me know if you want copies of them.

NB: Almost every Maths textbook I’ve ever seen contains at least some errors and I have noted a couple in both of the books I mention above. I expect this may well be due to proof reading/typesetting done people who are not proficient mathematicians as opposed to mistakes made by the authors themselves; sometimes we just have to be a little more forbearing when dealing with ‘technical’ textbooks.

 

[Locus]

Good morning Highlander.
The book I am using is the standard, official, book that is to be used by kids doing the leaving certificate.
There is a higher level maths for kids who want to pursue a scientific education further on and there is an ordinary level maths for kids who want to pursue other fields of study, say, humanities or arts. This one is the higher level.

The book contains far too many mistakes to be ignored, some of them seem to be conceptual, not just editors mistakes. It is quite detrimental for kids who are trying to score 100% and need to cover the whole book without a fail, as it takes an average of three hours to figure out a mistake, and there are 6 other subjects to cover as well, 4 of which sciences. I am not posting every single one of them in here, just the ones I am really stuck at, usually due to exhaustion.
Anyway, as I said before, I am trying to avoid mentioning names, because I'm not here on a smear campaign, just trying to cut study time so that all subjects are covered equally 100%. Spending three hours on every mistake does not allow for that.
While I do enjoy going through a maths book even without a specific purpouse, a different maths book would not help me right now, because this one I'm using covers exactly what is expected to come up at the leaving cert exam, while another one, from a different country (or time) might not cover exactly what is needed, but either less or maybe more (the old one from my time for example is a little different from the one in use now).
So that's what I meant.
But thank you for your offer of an alternative book. The title would suffice me. I will probably order it online and go through it at my own time, if for nothing else but to keep my maths unrusty for the next generation, given that I might have already failed this one.

I do intend to send the three esteemed professors who compliled this book an email with the long list of mistakes there are in this book, after I am done, in the hope that they will not put another year of kids through the **** they put mine. I will do so privately at first, hence I am trying to avoid names for now. I am giving them the benefit of the doubt, though I have to say, if my name was at the cover of a book, I would make sure there are zero mistakes in it or I'll be damned. And if an edition is not made, then it will be at Twitter, Department of Education.

 

[stapel]

Good morning Highlander.
The book I am using is the standard, official, book that is to be used by kids doing the leaving certificate.
There is a higher level maths for kids who want to pursue a scientific education further on and there is an ordinary level maths for kids who want to pursue other fields of study, say, humanities or arts. This one is the higher level.

The book contains far too many mistakes to be ignored, some of them seem to be conceptual, not just editors mistakes. It is quite detrimental for kids who are trying to score 100% and need to cover the whole book without a fail, as it takes an average of three hours to figure out a mistake, and there are 6 other subjects to cover as well, 4 of which sciences. I am not posting every single one of them in here, just the ones I am really stuck at, usually due to exhaustion.
Anyway, as I said before, I am trying to avoid mentioning names, because I'm not here on a smear campaign, just trying to cut study time so that all subjects are covered equally 100%. Spending three hours on every mistake does not allow for that.
While I do enjoy going through a maths book even without a specific purpouse, a different maths book would not help me right now, because this one I'm using covers exactly what is expected to come up at the leaving cert exam, while another one, from a different country (or time) might not cover exactly what is needed, but either less or maybe more (the old one from my time for example is a little different from the one in use now).
So that's what I meant.
But thank you for your offer of an alternative book. The title would suffice me. I will probably order it online and go through it at my own time, if for nothing else but to keep my maths unrusty for the next generation, given that I might have already failed this one.

I do intend to send the three esteemed professors who compliled this book an email with the long list of mistakes there are in this book, after I am done, in the hope that they will not put another year of kids through the **** they put mine. I will do so privately at first, hence I am trying to avoid names for now. I am giving them the benefit of the doubt, though I have to say, if my name was at the cover of a book, I would make sure there are zero mistakes in it or I'll be damned. And if an edition is not made, then it will be at Twitter, Department of Education.

I can't argue with you! (And blessings upon you, for being willing to collect the errors and forward them to the publisher!)

 

[mmm4444bot]

standard, official, book

Have you checked the publisher's web site for an errata list?

?
[imath]\;[/imath]

 

[Locus]

Have you checked the publisher's web site for an errata list?

?
[imath]\;[/imath]

No. I will now, didn't cross my mind before.

 

[The Highlander]

There is a higher level maths for kids who want to pursue a scientific education further on and there is an ordinary level maths for kids who want to pursue other fields of study, say, humanities or arts. This one is the higher level.

But thank you for your offer of an alternative book. The title would suffice me. I will probably order it online and go through it at my own time, if for nothing else but to keep my maths unrusty for the next generation, given that I might have already failed this one.

Hi @Locus,

Apologies for the delayed response; always a 'busy' time of year (?/?/?/? etc.) so this was my first chance to reply meaningfully.

Rather than just quote the Titles/Authors of the texts we use here, I have attached a few pages extracted from the digital copies of the textbooks that I have. That way you will have not only the information you requested but also a 'flavour' of what's in these texts and you can better decide then whether they would be of any use to you; I trust this 'approach' may be of interest?

However, I do note (from your post here) that you mention the Secant Ratio. I should point out, therefore, that we do not introduce (even by name*) the reciprocal trigonometric ratios in our high schools here so neither of these books will address them at all.

Hope that helps. ??

* I have never explored the etymological origins of the names of the reciprocal trigonometric ratios (perhaps I should? ?) but I have always thought that it would have been much more sensible/logical to have named the Secant as the Cosecant and vice versa; the Cotangent being an obvious choice for the name of the reciprocal of the Tangent ratio. The names chosen have caused (me, at least) confusion in the past! ?

 

[Locus]

Thank you very much for the books man. Merry Christmas and a happy New Year.

If Latin was a little less confusing, it would still be in use today. ?

 

[Locus]

Thank you very much for the books man. Merry Christmas and a happy New Year.

If Latin was a little less confusing, it would still be in use today. ?

But jokes aside, to be a good teacher of maths, one must delve into the history of maths, hence latin, arabic and ancient greek come into cosideration.

 

[Steven G]

Good morning/evening everyone.

I'm going through a book riddled with mistakes and that causes me to doubt my math skills first, then the book and then the whole discipline of mathematics, and im dad, so you can imagine what it does to leaving cert teenage daughter.
This particular exercise already cost me almost a full day, so please help.
Exercise goes as follows:
Prove that:
cos(45degrees + theta) -(cos45degrees - theta) = -squarerootof2sintheta

I tried the same thing a few times and i consulted various sources on and offline so finally I came to the decision that they have made a mistake, and the exercise should have, instead, been:

Prove that:
cos(45degrees + theta) - cos(45degrees - theta) = -squarerootof2sintheta

Am I wrong in making that assumption and is there a way where the original exercise is right?

(cos45degrees - theta)
cos45degrees is a real number between -1 and 1.
theta is an angle.
You can't subtract an angle from a real number!
Also, to know something is wrong simply plug in any value for theta and see if both sides are equal.

 

[The Highlander]

(cos45degrees - theta)
cos45degrees is a real number between -1 and 1.
theta is an angle.
You can't subtract an angle from a real number!
Also, to know something is wrong simply plug in any value for theta and see if both sides are equal.

Er, that's all true, of course, Steven, but did you read Dr.P's reply (at post #2) before you posted this? His post pretty much cleared up the matter without the need for any further attention. ???

 

[The Highlander]

But jokes aside, to be a good teacher of maths, one must delve into the history of maths, hence latin, arabic and ancient greek come into cosideration.

@Locus,

Indeed, but, having (now) checked on the derivation, it would appear that it just comes from the Latin "secare" (to cut) so there doesn't appear to be any good reason why Secant couldn't have been chosen as the name for \(\displaystyle \frac{1}{sin}\) as opposed to \(\displaystyle \frac{1}{cos}\) so just some dumbass that decided to do it the "inconvenient" way it appears. Arrrgghh.

I still think it would have been much more sensible/logical to have called the reciprocal of the Cosine ratio the Cosecant, (since both would then start with "Cos..." and make it easier to remember which is which! Doh.)

BTW, did you look at the private conversation I opened with you? Thought you might have responded. ??

 

[Dr.Peterson]

(cos45degrees - theta)
cos45degrees is a real number between -1 and 1.
theta is an angle.
You can't subtract an angle from a real number!
Also, to know something is wrong simply plug in any value for theta and see if both sides are equal.

That isn't really true; an angle in radians is just a number (a dimensionless ratio), just as a trig function's value is. And you can graph a function like [imath]\cos(x)-x[/imath].

However, I don't know of any particular reason to actually do this. And the numerical check is what counts.

Indeed, but, having (now) checked on the derivation, it would appear that it just comes from the Latin "secare" (to cut) so there doesn't appear to be any good reason why Secant couldn't have been chosen as the name for \(\displaystyle \frac{1}{sin}\) as opposed to \(\displaystyle \frac{1}{cos}\) so just some dumbass that decided to do it the "inconvenient" way it appears. Arrrgghh.

I still think it would have been much more sensible/logical to have called the reciprocal of the Cosine ratio the Cosecant, (since both would then start with "Cos..." and make it easier to remember which is which! Doh.)

No, there is a good reason "secant" means what it does. It's a length of a particular secant segment, which was initially called the "hypotenuse" (see here), from this figure:

​

Certainly there are other names that could have been used, and other functions that could have been given this name; but this was probably the segment they had reason to give a name to first, after sine and tangent, and secant was as good a name as anything else.

Then the corresponding ratios for the complement of the angle were given the "co-".