How to show the sum of these cosines is -1/2? (And find the sum of the sines?)

How do I do the last part of this question? (Show the sum of sines and cosines)I was tempted to try and do some trig manipulation to find the value of cos(6pi/7) and use the sum of a geometric series to find the sum of cos(2pi/7), cos(4pi/7), cos(6pi/7) and cos(8pi/7) but that would be a lot of work and I don't think that's how the question was intended to be done

You have to use the $\alpha$ they have given you and the equation in some way.

Maybe write $\displaystyle \alpha =e ^{i \frac{2 \pi}{7}}$ to make the calculations easier.

Could you post your working out to the previous parts of the question?

PS: kwl name

How is the sum of the cosines related to the expression

alpha + alpha^2 + alpha^4?

I remember doing this quite a while ago. The main idea to keep in mind is that the roots of unity are conjugates of the other rootsPosted from TSR Mobile

That was the first thought that came into my head when doing the question, but I didn't see anything to it. I know it's the real part of the alpha expression, but I don't see anything useful I can do just by knowing it's the real part.

Have you found A, B? In which case don't you know what

alpha + alpha^2 + alpha^4

equals?

What's a root of unity and a conjugate? lol, I'm guessing something to do with a complex number

I haven't actually been taught anything about the imaginary numbers all I know is that euler formula and that they are the square root of -1

I'm just trying to wing this question with my (basic) AS level math knowledge.

No offence, but this isn't really the sort of question you should be trying to "wing" with basic AS knowledge!

I don't see why I shouldn't be trying to 'wing' these questions, I need practice for my as core modules coming up. I can do most of the questions on the paper I found them on but it's the complex number thing that screws with me a bit sometimes, good thing that's not on C2

I'll probably pick up that Further Pure textbook in the summer and go through it.

I've learned Euler and De Moivre's out of my own interest, but I think that's about all I have in my toolkit for tackling complex numbers.

The pointr

The "point" is that Euler and De Moivre are more advanced bits of complex numbers. You should have covered what a conjugate is in the very first 'lesson' you did in complex numbers - basically if z is a complex number z = x + iy then the the conjugate is z* = x - iy (sometimes denoted by z with a bar over the top). If z is complex then both zz* and z + z* are real. The nth roots of unity are basically the n distinct numbers that satisfy z^n = 1.

I'm not saying you can't wing the question - what I'm saying is that you should be prepared to go back to the textbook and grab the basics that you need first

You shouldn't have to learn that, it's just the difference of two squares and then adding and subtracting

hang on, just a second. My email is not receiving the pictures i sent from my iphone for some reason

I was giving you some (helpful) context because you implied that you didn't know what a conjugate was!

I tell you what: I'll stop trying to be helpful now, and you can tell us what working you've done so far! Have you got the values of A and B now, in which case you should be able to answer RichE's question from earlier?

yep posted workings
sorry, didn't know I pissed you off

How do I do the last part of this question? (Show the sum of sines and cosines)

I was tempted to try and do some trig manipulation to find the value of cos(6pi/7) and use the sum of a geometric series to find the sum of cos(2pi/7), cos(4pi/7), cos(6pi/7) and cos(8pi/7) but that would be a lot of work and I don't think that's how the question was intended to be done