Question
Sat August 11, 2012 By:
 

who to find the value of cos 3degree

Expert Reply
Sun August 12, 2012
First, find sin (18) and sin (15). Then use these to find cos (18) and cos (15); and then finally you can easily find sin (3).
 
sin 18 degrees:

sin (72) = 2 sin (36) cos (36)
By the double angle formula for sines again, 2 sin (36) = 4 sin (18) cos (18); and by the double angle formula for cosines, cos (36) = 1 - 2 sin^2 (18). Thus, we have
sin (72) = (4 sin (18) cos (18)) (1 - 2 sin^2 (18))
cos (18) = 4 sin (18) cos (18) (1 - 2 sin^2 (18))
Divide both sides by cos (18):
1 = 4 sin (18) (1 - 2 sin^2 (18))

Let x = sin 18
1 = 4x(1 - 2x^2)
1 = 4x - 8x^3
8x^3 - 4x + 1 = 0
(2x - 1)(4x^2 + 2x - 1) = 0
So we get x = 1/2 as one solution, and the quadratic formula gives the other solutions as x = (-1 + ?5) / 4 and x = (-1 - ?5) / 4.

Now, we know that sin 18 has to be one of these values.

sin (18) can't be 1/2, because sin (0) = 0, sin (30) = 1/2, and sin x is an increasing function between 0 and 30.

sin (18) can't be (-1 - ?5) / 4, because sin x is positive when x is between 0 and 180.

Therefore, sin (18) = (-1 + ?5) / 4.
 
sin 15 degrees:
By the half-angle formula for sines, we have

sin (30/2) = ?{(1 - cos 30) / 2}

But cos (30) = ?3 / 2, so we have:

sin (15) = ?{(1 - ?3 / 2) / 2}
sin (15) = ?((2/2 - ?3 / 2) / 2}
sin (15) = ?{(2 - ?3) / 4}
sin (15) = ?(2 -?3) / 2.

It turns out that ?(2 -?3) is the same as (?6 - ?2) / 2. This is not obvious, but if you square (?6 - ?2) / 2 and simplify, it turns out that you get 2 -?3. So the answer can be simplified a bit further:

sin (15) = (?6 - ?2) / 4.
 
Now, use the values of sin (18) and sin (15) to find the values of cos (18) and cos (15). Make use of the identity sin^2 x + cos^2 x = 1.
 
You will get:
cos (18) = ?{(10 + 2?5)} / 4
cos (15) = (?6 + ?2) / 4
 
cos 3 degrees:
To find cos 3, use the identity
cos (A - B) = cos A cos B + sin A sin B
Put A = 18 and B = 15
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