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October 10, 2015
Posted by **Edwards** on Wednesday, August 3, 2011 at 9:27am.

Without using a calculator, show that è = 18 degrees is an exact solution of the equation Sin3è = Cos2è.

Justifying your answer, find the exact values of:

i) Sin18 degrees.

ii) Sin234 degrees.

- Mathematics -
**Reiny**, Wednesday, August 3, 2011 at 10:31amI will use x instead of è

sin 3x - cos 2x = (1-sinx)(4sin^2 x + 2sinx - 1)

RS = 4sin^2 x + 2sinx - 1 - 4sin^3 x + 2sin^2 x + sinx

= -4sin^3 x + 2sin^2 x + 3sinx - 1

LS = sin(2x+x) - cos 2x

= (sin 2x)(cosx) + (cos 2x)(sinx) - (1 - 2sin^2 x)

= 2sinxcosxcosx + sinx(1-2sin^2 x) - 1 + 2sin^2 x

= 2sinxcos^2 x + sinx - 2sin^3 x - 1 + 2sin^2 x

= 2sinx(1 - sin^2 x) + sinx - 2sin^3 x - 1 + 2sin^2 x

= 2sinx - 2sin^3 x + sinx - 2sin^3 x - 1 + 2sin^2 x

= -4sin^3 x + 2sin^2 x + 3sinx - 1

= RS

For your next question, show that 18° is a solution to

sin 3x = cos 2 or

sin 3x - cos 2x = 0

since I expanded this expression in LS above

we have to solve

-4sin^3 x + 2sin^2 x + 3sinx - 1 = 0 or

4sin^3 x - 2sin^2 x - 3sinx + 1 = 0

let sinx = y

4y^3 - 2y^2 - 3y + 1 = 0

clearly y = 1 is a solution

(y-1)(4y^2 + 2y -1) = 0

the other roots are (-1 ± √5)/4

so sinx = (-1 ± √5)/4

This also answers your second-last question.

that sin 18° = (√5 - 1)/4

I now have to show that this equals sin 18° without a calculator, (using my calculator shows me that so far I am correct, since 18 would be a solution using my machine)

Working on this ......

- Mathematics -
**Reiny**, Wednesday, August 3, 2011 at 11:24amsin 18° = cos 72° by the complementary property.

going out on a far-fetched limb here .......

draw a pentagon ABCDE

draw diagonals AC and BD to intersect at P

Look at triangle ABC, angle B = 108 and angles A and C are 36° each.

If AB = 1, I happen to know that the diagonal : side = the golden ratio

which is (1 + √5)/2 : 1

so let AB = 2, then AC = √5 + 1

using the cosine law in triangle ABC

2^2 = 2^2 + (√5+1)^2 – 2(2)(√5+1)cos 36°

4 = 4 + 5 + 2√5 + 1 – 4(√5+1)cos 36

cos 36 = (6+2√5)/(4(√5+1))

= (√5 + 1)/4 after rationalizing the denominator.

Now cos 72 = 2cos^2 36° - 1 , using cos 2A = 2cos^2 A - 1

= 2[(√5+1)/4]^2 – 1

= 2(5 + 2√5 + 1)/16 – 1

= (12 + 4√5)/16 – 16/16

= (-4 + 4√5)/16

= (-1 + √5)/4

but sin 18 = cos 72, as noted above

so sin 18° = (-1 + √5)/4

- Mathematics -
**Reiny**, Wednesday, August 3, 2011 at 11:33amfor your last one:

234 = 180 + 54

and 54 = half of 108 which is the large angle in the pentagon.

sin234 = sin(180 + 54) = sin180cos54 + cos180sin54

= 0 - sin54

= -sin 54°

you can find cos 108° using the cosine law in triangle ABC

then

cos 108 = 1 - 2sin^2 54°

give it a shot.