Answer:
- a = 24(97√3 -168) ≈ 0.21428002
- b = 144(26 -15√3) ≈ 2.77025565
- c = 144(97 -56√3) ≈ 0.74228776
Explanation:
We can consider the horizontal line to be y=0, and define p, q, r to be the points on the top line left to right between c and 1 at the vertices of the 15° angles.
Then the y-value of r is 1 -tan(15°)². The y-value of q is the square of this, and the y-value of p is the cube (1 -tan(15°)²)³. That makes the value of c be ...
c = (1 -tan(15°)²)⁴
15° is half of 30°, so we can use the tangent half-angle identity to find its value:
tan(15°) = (1 -cos(30°))/sin(30°) = (1 -√3/2)/(1/2) = 2 -√3
This lets us find the exact value of c to be ...
c = (1 -(2 -√3)²)⁴ = (1 -(4 -4√3 +3))⁴ = (4√3 -6)⁴
= (4√3)⁴ -4·6·(4√3)³ +6·6²·(4√3)² -4·6³·(4√3) +6⁴
= 2304 -4608√3 +10368 -3456√3 +1296 = 13968 -8064√3
c = 144(97 -56√3)
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The value of 'a' is ...
a = p·tan(15°) = (4√3 -6)³·(2 -√3) = (2 -√3)((4√3)³ -3·6·(4√3)² +3·6²(4√3) -6³)
= (2 -√3)(192√3 -864 +432√3 -216) = (2 -√3)(624√3 -1080)
= 1248√3 -2160 -1872 +1080√3 = 2328√3 -4032
a = 24(97√3 -168)
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The value of b is ...
b = c/tan(15°) = (144)(97 -56√3)/(2 -√3) = (144)(2 +√3)(97 -56√3)/(4 -3)
= 144(194 -112√3 +97√3 -168)/1 = 144(26 -15√3)
b = 144(26 -15√3)
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Additional comment
The tangent of an angle in a right triangle is the ratio of the opposite to the adjacent side. For example, tan(15°) = c/b. Then the right-most short segment of the horizontal line is 1·tan(15°). The difference between the vertical at its right (1) and the vertical at its left (r) = tan(15°) times that segment length, or tan(15°)². Then r = 1-tan(15°)². Each vertical segment to the left of that is the previous vertical segment multiplied by this factor:
q = r²
p = r·q = r³
c = r·p = r⁴
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The binomial expansions we used are ...
(a +b)⁴ = a⁴ +4a³b +6a²b² +4ab³ +b⁴
(a +b)³ = a³ +3a²b +3ab² +b³