Best answer:
Three points on the curve y = sinx:
A (a, sin a), B (b, sin b), C (c, sin c)
where 0 ≤ a < b < c ≤ 2π
Triangle ABC has base |AC| = √((a − c)² + (sin a − sin c)²)
Line AC has slope = (sin a − sin c) / (a − c)
Equation of line AC:
y − sin a = (sin a − sin c) / (a − c) (x − a)
(a − c) (y − sin a) = (sin a −...
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Best answer:
Three points on the curve y = sinx:
A (a, sin a), B (b, sin b), C (c, sin c)
where 0 ≤ a < b < c ≤ 2π
Triangle ABC has base |AC| = √((a − c)² + (sin a − sin c)²)
Line AC has slope = (sin a − sin c) / (a − c)
Equation of line AC:
y − sin a = (sin a − sin c) / (a − c) (x − a)
(a − c) (y − sin a) = (sin a − sin c) (x − a)
(sin a − sin c) (x − a) + (c − a) (y − sin a) = 0
Triangle ABC has height = distance from B to line AC
= |(sin a − sin c) (b − a) + (c − a) (sin b − sin a)| / √((sin a − sin c)² + (a − c)²)
= |(b−c) sin a + (c−a) sin b + (a−b) sin c| / √((sin a − sin c)² + (a − c)²)
Area (ABC)
= 1/2 * √((a − c)² + (sin a − sin c)²) * |(b−c) sin a + (c−a) sin b + (a−b) sin c| / √((sin a − sin c)² + (a − c)²)
= 1/2 * |(b−c) sin a + (c−a) sin b + (a−b) sin c|
Using WolframAlpha, we find area is maximized when:
a = 0
b = 1.72545
c = 5.32211
Maximum area = 3.33656
http://www.wolframalpha.com/input/?i=max...
Here is a diagram showing the triangle with maximum area:
https://www.desmos.com/calculator/di4j6s...
You can move the points around on the curve.
The area is shown on the second line.