1. Show that Sin3è - Cos2è = (1 - Sinè)(4Sin^2è + 2Sinè - 1).
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.
I 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 ......
sin 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
for 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.
To solve the first part of the problem, we need to express Sin3è and Cos2è in terms of Sinè.
1) Show that Sin3è - Cos2è = (1 - Sinè)(4Sin^2è + 2Sinè - 1).
First, let's find the values of Sin3è and Cos2è in terms of Sinè.
Knowing that Sin(A + B) = SinA * CosB + CosA * SinB, we can rewrite Sin3è as follows:
Sin3è = Sin(2è + è) = Sin2è * Cosè + Cos2è * Sinè.
Substituting Cos2è = 1 - 2Sin^2è, we get:
Sin3è = Sin2è * Cosè + (1 - 2Sin^2è) * Sinè.
Expressing Sin2è in terms of Sinè, we have:
Sin2è = 2Sinè * Cosè.
Substituting this back into the equation for Sin3è, we have:
Sin3è = 2Sinè * Cosè * Cosè + (1 - 2Sin^2è) * Sinè.
Simplifying this equation:
Sin3è = 2Sinè * Cos^2è + Sinè - 2Sin^3è.
To simplify further, factor out Sinè:
Sin3è = Sinè(2Cos^2è - 2Sin^2è) + Sinè(1 - 2Sin^2è).
Notice that 2Cos^2è - 2Sin^2è = 2(Cos^2è - Sin^2è) = 2Cos(2è).
Replacing 2Cos^2è - 2Sin^2è with 2Cos(2è), we get:
Sin3è = Sinè(2Cos(2è)) + Sinè(1 - 2Sin^2è).
Factoring Sinè out again:
Sin3è = Sinè(2Cos(2è) + 1 - 2Sin^2è).
Now, we know that 1 - Sinè = Cos(90 - è). Therefore, substituting this into our equation:
Sin3è = Sinè(2Cos(2è) + Cos(90 - è) - 2Sin^2è).
Using the double-angle formula, we can write 2Cos(2è) as 2(2Cos^2è - 1).
Substituting this into the equation:
Sin3è = Sinè(2(2Cos^2è - 1) + Cos(90 - è) - 2Sin^2è).
Expanding the equation:
Sin3è = Sinè(4Cos^2è - 2 + Cos(90 - è) - 2Sin^2è).
Using the identity CosA = Sin(90 - A), we can simplify the equation further:
Sin3è = Sinè(4Cos^2è - 2 + Sinè - 2Sin^2è).
Rearranging the terms:
Sin3è = Sinè(4Sin^2è + 2Sinè - 1).
Therefore, we have shown that Sin3è - Cos2è = (1 - Sinè)(4Sin^2è + 2Sinè - 1).
2) To prove that è = 18 degrees is an exact solution of the equation Sin3è = Cos2è, we can substitute è = 18 degrees into the equation and check if the equation holds true.
Let's substitute è = 18 degrees into the equation Sin3è = Cos2è:
Sin(3 * 18) = Cos(2 * 18).
Simplifying:
Sin54 = Cos36.
We know that Sin54 = Sin(90 - 36) = Sin54 = Cos36.
Therefore, Sin3è = Cos2è holds true for è = 18 degrees.
3) To find the exact values of Sin18 degrees and Sin234 degrees:
i) Sin 18 degrees:
We can use the angle addition formula for Sin(A + B) to express Sin18 degrees in terms of smaller angles.
Sin18 = Sin(9 + 9) = Sin(9)Cos(9) + Cos(9)Sin(9).
Using the half-angle formula for Sin(9), we know that:
Cos(18) = 2Cos^2(9) - 1.
Substituting this back into the equation:
Sin18 = Sin(9)(2Cos^2(9) - 1) + (1 - 2Sin^2(9))Sin(9).
Expanding and simplifying further:
Sin18 = 2Sin(9)Cos^2(9) - Sin(9) + Sin(9) - 2Sin^3(9).
Simplifying again:
Sin18 = 2Sin(9)Cos^2(9) - 2Sin^3(9).
We can use the half-angle formula for Cos(9) to express it in terms of Sin(9):
Cos(9) = Sqrt((1 + Cos(18)) / 2).
Plugging in the value of Cos(18) from step 2:
Cos(9) = Sqrt((1 + 2Cos^2(9) - 1) / 2) = Sqrt(Cos^2(9)) = Cos(9).
Replacing Cos^2(9) with Cos(9) in the Sin18 equation:
Sin18 = 2Sin(9)Cos(9) - 2Sin^3(9).
Now, we have reduced the problem to finding Sin(9). We can't find the exact value of Sin(9) without using a calculator, but we can apply numerical methods or use a calculator to approximate it.
ii) Sin234 degrees:
Sin234 can be expressed using periodicity of the sine function. Sin234 is equivalent to Sin(180 + 54). We know that Sin(180 + θ) = Sinθ.
Therefore, Sin234 = Sin54.
We already calculated the exact value of Sin54 in step 2, which is equal to Cos36.
So, Sin234 = Cos36.
Note that Cos36 can also be expressed using periodicity of the cosine function. Cos36 is equivalent to Cos(360 - 36). We know that Cos(360 - θ) = Cosθ.
Therefore, Sin234 = Cos36 = Cos(360 - 36) = Cos324.
Unfortunately, finding the exact value of Cos324 without using a calculator is not possible. However, we can use a calculator or other numerical methods to approximate its value.