Maths C3 - Trigonometry... Help??

OP

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 :smile:

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 :smile:

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!

Reply 262

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" :smile:

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.

(edited 7 years ago)

Reply 263

OP

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" :smile:

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!! :smile:

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.

Reply 264

OP

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 :frown:

Reply 265

B_9710

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 :frown:

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'.

Reply 266

IrrationalRoot

20

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 :frown:

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.

Reply 267

OP

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 :frown:

Any more info would be more than appreciated :smile:

Reply 268

B_9710

https://en.m.wikipedia.org/wiki/Even_and_odd_functions

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 :frown:

Any more info would be more than appreciated :smile:

Reply 269

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 :frown:

Any more info would be more than appreciated :smile:

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.

Reply 270

IrrationalRoot

20

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 :frown:

Any more info would be more than appreciated :smile:

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.

Reply 271

OP

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 :smile:

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!! :smile:

Reply 272

OP

Ok so I'm stuck on Q5(h)...

C3 Exe7A Q5.png

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 :frown:

Reply 273

Original post by Philip-flop

Ok so I'm stuck on Q5(h)...

C3 Exe7A Q5.png

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 :frown:

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.

Reply 274

OP

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}

?

Reply 275

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.

Reply 276

OP

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 :smile:

Is a bit of a weird Q though :s-smilie:

Reply 277