1) for P, how far away is it from the center? You need the distance to N for the x coordinate, hence cos(A), Just remember SOHCAHTOA. For the y coordinate, you need the distance to P, which will be sin(A).(Original post by Philipflop)
So I've just started Chapter 7 of the Edexcel C3 Modular Maths textbook and have already encountered a problem
It's about proving the 'addition formula' for cos(AB) = cosAcosB + sinAsinB
Attachment 583362
Ok so I have the following questions to ask about this example...
 Why are the coordinates of P and Q... P(cosA, sinA) and Q(cosB, sinB)?
 Why is the radius of the circle 1?
 Why is the angle POQ equal to (AB)?? Surely it would be angle POQ = (BA), is it because using the addition formulae cos(ab) = cos(ba)??
 How would one know to compare the lengths of PQ^2 from the results of using the cosine rule and the distance between two coordinates for this type of question?
2) Same thing for Q.
3) Radius of the circle is 1 either because the question will mention something about the radius, or it simplifies the problem.
4) Idk why it's AB
5) You have to show that cos(AB) is cosAcosB + sinAsinB, and from the length of PQ^2 you get an answer and from cosine rule you get another answer, they must be the same, because they both in the form PQ^2 =

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 29092016 13:29

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 29092016 13:30
(Original post by Philipflop)
So I've just started Chapter 7 of the Edexcel C3 Modular Maths textbook and have already encountered a problem
It's about proving the 'addition formula' for cos(AB) = cosAcosB + sinAsinB
Attachment 583362
Ok so I have the following questions to ask about this example...
 Why are the coordinates of P and Q... P(cosA, sinA) and Q(cosB, sinB)?
 Why is the radius of the circle 1?
 Why is the angle POQ equal to (AB)?? Surely it would be angle POQ = (BA), is it because using the addition formulae cos(ab) = cos(ba)??
 How would one know to compare the lengths of PQ^2 from the results of using the cosine rule and the distance between two coordinates for this type of question?
For the coordinates of P, the xcoordinate is the horizontal distance from O which is the distance ON and similarly the ycoordinate is the distance NP. Both of these can be found using SOHCAHTOA with angle = A and hypotenuse = 1.
The coordinates of Q can be found in a similar way.
They should have used POQ = B  A. But in the end it doesn't matter because cos(A) = cos(A) so cos(AB) = cos(BA).
Regarding your last point, this isn't really a "type of question". This is just one way you can prove the addition formula for cos. You would not be expected to prove something like this in the exam from scratch without more guidance. 
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 29092016 13:54
(Original post by asinghj)
1) for P, how far away is it from the center? You need the distance to N for the x coordinate, hence cos(A), Just remember SOHCAHTOA. For the y coordinate, you need the distance to P, which will be sin(A).
2) Same thing for Q.
3) Radius of the circle is 1 either because the question will mention something about the radius, or it simplifies the problem.
4) Idk why it's AB
5) You have to show that cos(AB) is cosAcosB + sinAsinB, and from the length of PQ^2 you get an answer and from cosine rule you get another answer, they must be the same, because they both in the form PQ^2 =
(Original post by notnek)
Firstly, you could choose the radius of the circle to be any positive value and this proof would still work. A radius of 1 makes things simple. You could even use a variable to denote the radius but it will make the algebra harder.
For the coordinates of P, the xcoordinate is the horizontal distance from O which is the distance ON and similarly the ycoordinate is the distance NP. Both of these can be found using SOHCAHTOA with angle = A and hypotenuse = 1.
The coordinates of Q can be found in a similar way.
They should have used POQ = B  A. But in the end it doesn't matter because cos(A) = cos(A) so cos(AB) = cos(BA).
Regarding your last point, this isn't really a "type of question". This is just one way you can prove the addition formula for cos. You would not be expected to prove something like this in the exam from scratch without more guidance.
Also, I'm glad to hear that they will give more guidance for the actual exam! I tried so hard to work this out without following the example at first and had no idea what to do! 
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 29092016 14:02
(Original post by Philipflop)
Also, I'm glad to hear that they will give more guidance for the actual exam! I tried so hard to work this out without following the example at first and had no idea what to do!
I know that it seems like this will be a common C3 question when the textbook puts it in "Example 1"
Of course you may be expected to prove e.g. the tan(A + B) identitiy using the sin(A + B) and cos(A + B) identities. Alegbraic proofs like this are common.Last edited by Notnek; 29092016 at 14:04. 
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 29092016 14:19
(Original post by notnek)
Geometric proofs of the addition formulae are not in the Edexcel C3 spec and I've never seen a past paper questions relating to them. That doesn't they won't come up but as I said before, you won't be expected to come up with the proof from scratch.
I know that it seems like this will be a common C3 question when the textbook puts it in "Example 1"
Of course you may be expected to prove e.g. the tan(A + B) identitiy using the sin(A + B) and cos(A + B) identities. Alegbraic proofs like this are common.
You have no idea how reassuring that is for a selfstudier like me! Once I start doing past papers I will get a grasp of the kind of questions that come up about the addition formulae.
Yeah proving tan(A+B) seems more straight forward to me than the one that was given in Example 1. 
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 29092016 14:47
Silly little question but...
how does...
and...
??
I was told this once or twice before but don't fully understand why 
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 29092016 14:54
(Original post by Philipflop)
Silly little question but...
how does...
and...
??
I was told this once or twice before but don't fully understand why
Because of this we say that cosine is an 'even function' and some is an 'odd function'. 
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 29092016 15:06
(Original post by Philipflop)
Silly little question but...
how does...
and...
??
I was told this once or twice before but don't fully understand why
I can guarantee that the vast majority of ALevel students don't even understand what the trig functions are. Look up 'unit circle definition of trig functions' for a definition and ask if there's anything you don't understand. If you use the actual definition of the trig functions, you will have a complete understanding of them and will find a lot of questions far easier than before.
Hopefully after that, you'll easily see why the two identities you've given hold and you won't have to memorise them like everyone else. 
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 29092016 16:34
(Original post by B_9710)
If you look at the curves then this relationship is easy to see.
Because of this we say that cosine is an 'even function' and some is an 'odd function'.
(Original post by IrrationalRoot)
It's not a silly question. It should be trivial to prove but because trigonometry is not taught properly at all in schools, no one can prove it (graphs are certainly not proofs).
I can guarantee that the vast majority of ALevel students don't even understand what the trig functions are. Look up 'unit circle definition of trig functions' for a definition and ask if there's anything you don't understand. If you use the actual definition of the trig functions, you will have a complete understanding of them and will find a lot of questions far easier than before.
Hopefully after that, you'll easily see why the two identities you've given hold and you won't have to memorise them like everyone else.
“Intro to Unit Circles”https://www.youtube.com/watch?v=1m9p9iubMLU“Unit Circle Definition of Trig Functions”https://www.youtube.com/watch?v=ZffZvSH285c
Unit circles are beginning to become more clearer now but I've still got a lot to learn
Any more info would be more than appreciated 
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 29092016 16:54
(Original post by Philipflop)
Can you tell me a little more about 'even' and 'odd' functions?
So I've just watched the following videos...
“Intro to Unit Circles”https://www.youtube.com/watch?v=1m9p9iubMLU“Unit Circle Definition of Trig Functions”https://www.youtube.com/watch?v=ZffZvSH285c
Unit circles are beginning to become more clearer now but I've still got a lot to learn
Any more info would be more than appreciated 
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 29092016 17:04
(Original post by Philipflop)
Can you tell me a little more about 'even' and 'odd' functions?
So I've just watched the following videos...
“Intro to Unit Circles”https://www.youtube.com/watch?v=1m9p9iubMLU“Unit Circle Definition of Trig Functions”https://www.youtube.com/watch?v=ZffZvSH285c
Unit circles are beginning to become more clearer now but I've still got a lot to learn
Any more info would be more than appreciated
This means that if you plug e.g. 3 into an even function, the output will be the same as if you plug in 3. Or in general, inputting x gives you the same as inputting x i.e. f(x) = f(x). The diagram above shows this.
If you draw the cosine graph for positive and negative x then you'll see that it satisfies this property so e.g. is equal to .
Odd functions satisfy f(x) = f(x). Draw the sine graph and see if you can see why it satisfies this. 
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 29092016 18:04
(Original post by Philipflop)
Can you tell me a little more about 'even' and 'odd' functions?
So I've just watched the following videos...
“Intro to Unit Circles”https://www.youtube.com/watch?v=1m9p9iubMLU“Unit Circle Definition of Trig Functions”https://www.youtube.com/watch?v=ZffZvSH285c
Unit circles are beginning to become more clearer now but I've still got a lot to learn
Any more info would be more than appreciated
Consider the circle with radius 1 centred at the origin. Start at the point (1,0). The definition of is the xcoordinate of the point that results when you rotate the point (1,0) through an angle anticlockwise. is the ycoordinate of the point that results when you rotate the point (1,0) through an angle anticlockwise.
It is very helpful to draw the diagram out so you know what's going on. Notice how for , what you're doing is rotating clockwise through an angle because of the minus sign (opposite direction if you like). Now the point corresponding to is clearly directly above the point corresponding to . Therefore the ycoordinates of the points, that is, the sines of the angles, are opposite in sign, so you get and having opposite signs. The same logic can be applied to get the identity for cosine. 
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 29092016 20:17
(Original post by notnek)
If you draw the graph.......
(Original post by IrrationalRoot)
Ok so basically this is how the definition of trig functions goes..... 
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 29092016 20:19
Ok so I'm stuck on Q5(h)...
I noticed the hint in the book which states "tan(45) = 1"
I've tried replacing 1 with tan(45) but then it all goes downhill for me from there 
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 29092016 20:23
(Original post by Philipflop)
Ok so I'm stuck on Q5(h)...
I noticed the hint in the book which states "tan(45) = 1"
I've tried replacing 1 with tan(45) but then it all goes downhill for me from there
Hint : Only replace the '1' on the numerator with tan(45). Leave the '1' on the denominator alone.
Then look at the tan(A + B) identity and see how it compares. 
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 29092016 20:32
(Original post by notnek)
Yes this is quite a sneaky one.
Hint : Only replace the '1' on the numerator with tan(45). Leave the '1' on the denominator alone.
Then look at the tan(A + B) identity and see how it compares.
but for me to use 'addition formulae' I need to get that equation in the form...
So will I just have to times the tan(15) in the numerator by tan(45)? to give me...
???
Which then leads to the answer being...
? 
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 29092016 20:34
(Original post by Philipflop)
ok so if I do that I get...
but for me to use 'addition formulae' I need to get that equation in the form...
So will I just have to times the tan(15) in the numerator by tan(45)? to give me...
???
Which then leads to the answer being...
?
This question always catches students out. You did it faster than most. 
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 29092016 20:37
(Original post by notnek)
Correct
This question always catches students out. You did it faster than most.
Is a bit of a weird Q though 
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 29092016 20:40
(Original post by Philipflop)
Yayyy. Glad I'm not the only person who gets caught out on this question then
Is a bit of a weird Q though
But the trick makes it a lot quicker. And the question writers deliberately put in (g) before (h) to give you a hint. 
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 29092016 21:08
Expanded as in like?....
Where do I go from here though?
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Updated: April 23, 2017
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