Maths C3 - Trigonometry... Help??

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    (Original post by Philip-flop)
    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(A-B) = cosAcosB + sinAsinB
    Attachment 583362

    Ok so I have the following questions to ask about this example...
    1. Why are the coordinates of P and Q... P(cosA, sinA) and Q(cosB, sinB)?
    2. Why is the radius of the circle 1?
    3. Why is the angle POQ equal to (A-B)?? Surely it would be angle POQ = (B-A), is it because using the addition formulae cos(a-b) = cos(b-a)??
    4. 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?
    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 A-B

    5) You have to show that cos(A-B) 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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    (Original post by Philip-flop)
    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(A-B) = cosAcosB + sinAsinB
    Attachment 583362

    Ok so I have the following questions to ask about this example...
    1. Why are the coordinates of P and Q... P(cosA, sinA) and Q(cosB, sinB)?
    2. Why is the radius of the circle 1?
    3. Why is the angle POQ equal to (A-B)?? Surely it would be angle POQ = (B-A), is it because using the addition formulae cos(a-b) = cos(b-a)??
    4. 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?
    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 x-coordinate is the horizontal distance from O which is the distance ON and similarly the y-coordinate 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(A-B) = cos(B-A).

    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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    (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 A-B

    5) You have to show that cos(A-B) 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 =
    Thank you so much asinghj That has cleared up so much for me!!

    (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 x-coordinate is the horizontal distance from O which is the distance ON and similarly the y-coordinate 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(A-B) = cos(B-A).

    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.
    Thanks notnek for the detailed response in how to work out the co-ordinates for P using SOHCAHTOA

    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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    (Original post by Philip-flop)
    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!
    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.
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    (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.
    Thank you!!
    You have no idea how reassuring that is for a self-studier 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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    Silly little question but...

    how does...  cos(-x) = cos(x)
    and...
     sin(-x) = -sinx ??

    I was told this once or twice before but don't fully understand why
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    (Original post by Philip-flop)
    Silly little question but...

    how does...  cos(-x) = cos(x)
    and...
     sin(-x) = -sinx ??

    I was told this once or twice before but don't fully understand why
    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'.
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    (Original post by Philip-flop)
    Silly little question but...

    how does...  cos(-x) = cos(x)
    and...
     sin(-x) = -sinx ??

    I was told this once or twice before but don't fully understand why
    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 A-Level 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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    (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'.
    Can you tell me a little more about 'even' and 'odd' functions?

    (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 A-Level 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.
    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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    (Original post by Philip-flop)
    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
    https://en.m.wikipedia.org/wiki/Even_and_odd_functions
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    (Original post by Philip-flop)
    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
    If you draw the graph of an even function then the stuff to the right of the y-axis is an exact reflection of the stuff to the left



    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 \cos(x) = \cos(-x) e.g. \cos(-34) is equal to \cos(34).

    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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    (Original post by Philip-flop)
    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
    Ok so basically this is how the definition of trig functions goes.

    Consider the circle with radius 1 centred at the origin. Start at the point (1,0). The definition of \cos\theta is the x-coordinate of the point that results when you rotate the point (1,0) through an angle \theta anticlockwise. \sin\theta is the y-coordinate of the point that results when you rotate the point (1,0) through an angle \theta anticlockwise.

    It is very helpful to draw the diagram out so you know what's going on. Notice how for \sin(-\theta), what you're doing is rotating clockwise through an angle \theta because of the minus sign (opposite direction if you like). Now the point corresponding to \theta is clearly directly above the point corresponding to -\theta. Therefore the y-coordinates of the points, that is, the sines of the angles, are opposite in sign, so you get \sin\theta and \sin(-\theta) having opposite signs. The same logic can be applied to get the identity for cosine.
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    (Original post by notnek)
    If you draw the graph.......
    Thank you for your reply with visuals! 'Even' and 'Odd' functions definitely make more sense to me now

    (Original post by IrrationalRoot)
    Ok so basically this is how the definition of trig functions goes.....
    Thanks a lot. That has definitely helped me out a lot!!
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    Ok so I'm stuck on Q5(h)...

    Name:  C3 Exe7A Q5.png
Views: 14
Size:  10.5 KB

    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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    (Original post by Philip-flop)
    Ok so I'm stuck on Q5(h)...

    Name:  C3 Exe7A Q5.png
Views: 14
Size:  10.5 KB

    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
    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.
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    (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.
    ok so if I do that I get...

     \frac{tan(45)-tan(15)}{1+tan(15)}

    but for me to use 'addition formulae' I need to get that equation in the form...
     \frac{tan A - tan B}{1+tan A tan B}

    So will I just have to times the tan(15) in the numerator by tan(45)? to give me...
     \frac{tan(45)-tan(15)}{1+tan(15)tan(45)} ???

    Which then leads to the answer being...
     \frac{\sqrt 3}{3} ?
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    (Original post by Philip-flop)
    ok so if I do that I get...

     \frac{tan(45)-tan(15)}{1+tan(15)}

    but for me to use 'addition formulae' I need to get that equation in the form...
     \frac{tan A - tan B}{1+tan A tan B}

    So will I just have to times the tan(15) in the numerator by tan(45)? to give me...
     \frac{tan(45)-tan(15)}{1+tan(15)tan(45)} ???

    Which then leads to the answer being...
     \frac{\sqrt 3}{3} ?
    Correct

    This question always catches students out. You did it faster than most.
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    (Original post by notnek)
    Correct

    This question always catches students out. You did it faster than most.
    Yayyy. Glad I'm not the only person who gets caught out on this question then
    Is a bit of a weird Q though
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    (Original post by Philip-flop)
    Yayyy. Glad I'm not the only person who gets caught out on this question then
    Is a bit of a weird Q though
    Of course you could have written both tan(15)'s as tan(60-45) and expanded them.

    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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    Expanded as in like?....
    Name:  Photo 29-09-2016, 21 06 59.jpg
Views: 25
Size:  97.4 KB

    Where do I go from here though?
 
 
 
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