Complex Numbers Question (1 Viewer)

[Ozil11]

Hi guys, just wondering if anyone would lend a hand,
1. Show that cos2pi/5 + cos4pi/5 = -1/2
2. Hence find the exact value of cos36

Thanks in advance

 

[asianese]

Notice that the argument of cosine has a factor of 1/5. This suggests the 5th roots of unity. Start from this.

 

[Drongoski]

I recall in one school's test, there was a similar question; show that:





etc

 

[integral95]

For part 2

It's using double angle formulas

so Cos 4pi/5 = 2(cos 2pi/5)^2 - 1

Now let x = cos 2pi/5

Therefore you would get x + 2x^2 - 1 = -1/2 (Using part 1)

Rearrange that to get 4x^2 + 2x - 1 = 0

We know that one of the solution is x = cos 2pi/5 (cos 72)

Using the quadratic formula

x = cos 72 = (-1+ squareroot5)/4 (note that cos 72 is positive so I only took the positive solution

Now use double angle formula again

2(cos36)^2 - 1 = (-1+ squareroot5)/4

From there you should get your answer

I need to learn LAtex lol

 

[integral95]

I recall in a King's School test, there was a similar question; show that:

Kings always make you do extra stuff like that (in this case you have to realise that cos(4pi/5) = -cos(pi/5)

 

[Ozil11]

wow thanks guys