Proving sin18 = √5-1/4 without Calculator: Double-Angle Formulae Method

[Radwa Kamal]

HELP! trigonometry

Homework Statement

without using calculator:
prove that: sin18 = √5-1/4

Homework Equations

Double-angle formulae
sin 2a = 2sin a.cos a
cos 2a = cos^2a - sin^2a = 1 - 2sin^2a = 2cos^2a - 1
sin^2 a + cos^2 a=1

The Attempt at a Solution

let 18=a
sin a = sin(90 - a) = cos 4a
cos 4a = 1 - 2sin^2 2a
= 1 - 2(2sin a.cos a)^2
= 1 - 2(4sin^2 a.cos^2 a)
= 1 - 8sin^2 a.cos^2 a
= 1 - 8sin^2a(1-sin^2a)
= 1 - 8sin^2a + 8sin^4a
i can't go further than that ?!

 

[Mentallic]

So now what you have is:

$$sina=1-8sin^2a+8sin^4a$$

Hint: To solve this quartic, search for rational factors with the rational factor theorem.