Question

Cos (pi/5) + cos (2pi/5)+ Cos (3pi/5)
+ Cos (4pi/5)

Answer 1

0

Keys:

When going over functions like this, we must use these cosine rules:

• 
  `\cos \left(s\right)+\cos \left(t\right)=2\cos \left((s+t)/(2)\right)\cos \left((s-t)/(2)\right)`
• 
  `\cos \left(-x\right)=\cos \left(x\right)`
• 
  `\cos \left((\pi )/(2)\right)=0`

Explanation:

`=\cos \left((\pi )/(5)\right)+2\cos \left((2\cdot (\pi )/(5)+3\cdot (\pi )/(5))/(2)\right)\cos \left((2\cdot (\pi )/(5)-3\cdot (\pi )/(5))/(2)\right)+\cos \left(4\cdot (\pi )/(5)\right)\\=\cos \left((\pi )/(5)\right)+2\cos \left((\pi )/(2)\right)\cos \left(-(\pi )/(10)\right)+\cos \left((4\pi )/(5)\right)\\=\cos \left((\pi )/(5)\right)+2\cos \left((\pi )/(2)\right)\cos \left((\pi )/(10)\right)+\cos \left((4\pi )/(5)\right)`

`cos\left((\pi )/(5)\right) = (√(5) + 1)/(4)\\=(√(5)+1)/(4)+2\cdot \:0\cdot \frac{√(2)\sqrt{5+√(5)}}{4}-(1+√(5))/(4)\\=0`

Answer 2

0

by using the cosine rule you can find this