## cos18°

During lessons about goniometry is explained how to calculate

the sine, cosine and tangent of 30°, 45° and 60°.

Can an exact calculation be done for other angles?

In the trangle below it is easily seen that 5α equals 90°.

Then α = 18°, 2α = 36° and 3α = 54°.

cos3α

From goniometry it is known, that cos3α = 4cos^{3}α - 3cosα.

You can find the proot below. You may skip it.You can see from the figure:The proof of cos3α = 4cos2 α - 3cosα

A standard formula from trigonometry is

cos(α + β) = cosαcosβ – sinαsinβ.

Choose β = 2α. Then

cos(α + 2α) = cosαcos2α – sinαsin2α

cos3α =

cosα(2cos^{2}α - 1) - sinα(2sinαcosα) =

cosα{(2cos^{2}α - 1) - 2sin^{2}α} =

cosα{(2cos^{2}α - 1) - 2(1 - cos^{2}α)}

So cos3α = 4cos^{3}α - 3cosα.cos3α =Here the Pythagorean theorem givesa/1. It is proven, that cos3α = 4cos^{3}α - 3cosα.

Soa= 4cos^{3}α - 3cosα. This is equation (1).

cos2α =b/1. It is known, that cos2α = 2cos^{2}α - 1,

Sob= 2cos^{2}α - 1. This is equation (2).a^{2}+b^{2}= 1.

Combinee it with equations (1) and (2):

(4cos^{3}α - 3cosα)^{2}+ (2cos^{2}α - 1)^{2}= 1

Temporarily change cosα intop.

(4p^{3}- 3p)^{2}+ (2p^{2}- 1)^{2}= 1

(16p^{6}- 24p^{4}+ 9p^{2}) + (4p^{4}- 4p^{2}+ 1) = 1

16p^{6}- 20p^{4}+ 5p^{2}= 0 ofp^{2}(16p^{4}- 20p^{2}+ 5) = 0.

p= 0 does not satisfy, so 16p^{4}- 20p^{2}+ 5 = 0.

Temporarily changep^{2}intox. Because 0 ≤ cosα ≤ 1, also 0 ≤p≤ 1 is true.

16x^{2}- 20x+ 5 = 0. It givesx= {5 + √5)/8.

p^{2}=x= (5 ± √5)/8 andp= cos18°, sop= √{(5 ± √5)/8}.

p^{2}= 0·9510565 orp= 0·5877852. According to my pocket calculator cos18° equals 0.9510565.

Apparently the exact description is:cos18° = √(5+√5)/8.

By means of the earlier mentioned formula cos2α = 2cos^{2}α - 1 it is possible to find

an exact expression for cos36° and cos72°. Even for cos9°.

With cos3α = 4cos2α - 3cosα we find an exact expression for cos54°.

With cos54° = 2cos^{2}27° – 1 we conjure up cos27°.

Sine

An exact expression of the sine of every mentioned angle can be found with cosα = sin(90° - α).

Tangent

An exact expression of the tangent of every mentioned angle can be found with tanα = sinα / cosα.

Look at the goniometrical table.

Another triangle

Using a triangle with angels α, 4α en 90° gives a similar result.

Multiples of 9°

With sin(α ± β) = sinαcosβ ± cosαsinβ and

cos(α + β) = cosαcosβ + sinαsinβ

all angels being a multiple of 9° can be found.

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