Thread: Finding cos (pi / 5)

1. Finding cos (pi / 5)

Hello. Sorry to be a nuisance to you guys but you've all been very helpful. This is going to be my last question.

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If:

sin(3pi / 10) = (1 + squareroot 5) / 4

Find the exact value of cos(pi / 5)

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This is the second part of a question. The first part mentioned something about sin(2pi / t) = cos t so I guess I have to somehow incorporate this identity. You don't have to solve it (although you could if you want to) but it would be great if you could provide me a starting point so I could work my way through. I was thinking that perhaps a unit circle diagram may help.

Thankyou very much. Of course, all help is appreciated. (happy)

2. Originally Posted by sqleung
Hello. Sorry to be a nuisance to you guys but you've all been very helpful. This is going to be my last question.

-----

If:

sin(3pi / 10) = (1 + squareroot 5) / 4

Find the exact value of cos(pi / 5)

[snip]
1. Note that $\displaystyle \frac{\pi}{2} - \frac{3 \pi}{10} = \frac{\pi}{5}$.

2. Complementary angle formula: $\displaystyle \cos \left( \frac{\pi}{2} - A \right) = \sin A$.

So substitute $\displaystyle A = \frac{3\pi}{10}$ into the above complemenatry angle formula.

3. Thankyou very much! With your information, I managed to get this:

Cos (pi / 5) = Cos((pi/2) - (3pi / 10))
Cos (pi / 5) = Cos(pi/2)Cos(3pi/10) + Sin(pi/2)Sin(3pi/10)
Cos (pi / 5) = Sin(3pi/10)
Cos (pi / 5) = (1 + squareroot 5)/4

Is that right?

4. Originally Posted by sqleung
Thankyou very much! With your information, I managed to get this:

Cos (pi / 5) = Cos((pi/2) - (3pi / 10))
Cos (pi / 5) = Cos(pi/2)Cos(3pi/10) + Sin(pi/2)Sin(3pi/10)
Cos (pi / 5) = Sin(3pi/10)
Cos (pi / 5) = (1 + squareroot 5)/4

Is that right?

(You can of course confirm it with a calculator by looking at appropriate decimal approximations of each side)

5. Thanks!

6. cos (pi/5)

Originally Posted by sqleung
Hello. Sorry to be a nuisance to you guys but you've all been very helpful. This is going to be my last question.

-----

If:

sin(3pi / 10) = (1 + squareroot 5) / 4

Find the exact value of cos(pi / 5)

-----

This is the second part of a question. The first part mentioned something about sin(2pi / t) = cos t so I guess I have to somehow incorporate this identity. You don't have to solve it (although you could if you want to) but it would be great if you could provide me a starting point so I could work my way through. I was thinking that perhaps a unit circle diagram may help.

Thankyou very much. Of course, all help is appreciated. (happy)
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Direction to solution:
Firstly, look at the roots of the equation z^5 + 1 = 0, where z is a complex number. This equation has 5 complex roots, where /z/=1 and arg(z) = (pi/5) + k*2(pi/5), with k element of .

The equation can be written in two fashions:
1) (z-1)*(z^2-2zcos(pi/5)+1)*(z^2-2zcos(3pi/5)+1)=0
2) (z-1)*(z^4 - z^3 + z^2 - z+1) = 0

These equations should be identical i.e. the coefficients are the same. This will yield two equations with cos(pi/5) and cos(3pi/5) the two unknowns.

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Value of cos(pie/5)

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