prove dat Cos(18°) = 1/4√10+2√5
Take a look at
http://mathworld.wolfram.com/TrigonometryAnglesPi10.html
There they use the half-angle formulas to evaluate cos(pi/10) in terms of cos(pi/5)
There is a nice discussion on evaluating cos(pi/5) at
https://socratic.org/questions/how-do-i-evaluate-cos-pi-5-without-using-a-calculator
Construct an isosceles triangle ABC
with angle A = 36°
AB = AC = 1
which leads to angle B = angle C = 72°
Let BC = x
Draw BD to hit AC at B, so that angle CBD = 36°
This makes BC = BD = x and
it makes triangle ABC similar to triangle BCD
Also triangle ABD is now isosceles, making AD = x , nice!
But AC = 1, so DC = 1-x
using triange ABC and triangle BCD
x/1 = (1-x)/x
x^2 = 1-x
x^2 + x - 1 = 0
x = (-1 ± √5)/2 , but x has to be positive
x = (√5 - 1)/2
using the cosine law on triangle ABC
[ (√5 - 1)/2 ]^2 = 1^2 + 1^2 - 2(1)(1)cos36°
(5 - 2√5 + 1)/4 = 2 - 2cos36
2cos36° = 2 - (3 - √5)/2
2cos36° = (4 - 3 + √5)/2
cos36° = (1 + √5)/4 <----- which just happens to be 1/2 the golden ratio!!!
recall cos2A = 1 - 2sin^2 A -----> cos36° = 1 - 2sin^2 18°
I will leave it up to you to to find sin 18° from that.
To prove that cos(18°) is equal to 1/(4√10) + 2√5, we can use the trigonometric identity for the cosine of a multiple of 18°.
The trigonometric identity that will help us here is:
cos(2x) = 2cos^2(x) - 1
Let's rewrite the given expression in terms of cos(18°):
1/(4√10) + 2√5 = (1/4)(1/√10) + 2√5
We can rationalize the denominator of the first term by multiplying the numerator and denominator by √10:
(1/4)(1/√10) = (1/4)(√10/√10) = √10/40
Now, let's substitute √10/40 back into our expression:
√10/40 + 2√5
Since our trigonometric identity involves doubling the angle, we can rewrite 18° as 2 * 9°.
Now, let's apply the trigonometric identity by substituting x = 9° into cos(2x):
cos(2 * 9°) = 2cos^2(9°) - 1
Let's represent cos(9°) as x. Now our equation becomes:
cos(18°) = 2x^2 - 1
Using a calculator or trigonometric table, we find that cos(9°) is approximately 0.9877.
Substituting this value into the equation:
cos(18°) = 2(0.9877)^2 - 1
cos(18°) ≈ 2(0.9755) - 1
cos(18°) ≈ 1.951 - 1
cos(18°) ≈ 0.951
Now, let's compare this result to the given expression of 1/(4√10) + 2√5:
0.951 ≈ 1/(4√10) + 2√5
Using decimal approximations for the radicals:
0.951 ≈ 1/(4 * 3.1623) + 2 * 2.2361
Simplifying the expression:
0.951 ≈ 1/12.6492 + 4.4722
To add these two fractions, we need to find a common denominator:
0.951 ≈ 1/(12.6492) + (4.4722 * 12.6492)/(12.6492)
0.951 ≈ 0.0791 + 56.6398
0.951 ≈ 56.7189
Since 0.951 is not approximately equal to 56.7189, the given equation 1/(4√10) + 2√5 is not equal to cos(18°). Thus, the statement is not valid.