Sides of a Triangle are in Arithmetic Progression
Math Problem: Sides of a Triangle are in arithmetic progression, then find the sines of angles
If the sides of the right-angle triangle are in arithmetic progression, then what are the sines of angles
Solution
Sides of the triangle are in arithmetic progression, so assume the sides of the triangles are x − a, x and x + a
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Now apply pythegorus therem in triangle, then
AC² = AB² + BC²
⇒ (x + a)² = x² + (x − a)²
⇒ x² + 2ax + a² = x² + x² − 2ax + a²
⇒ x² = 4ax
⇒ x = 4a
From triangle
sin A = BC / AC = x / (x + a) = 4a / (4a + a)
⇒ sin A = 4/5
sin B = sin 90°
⇒ sin B = 1
sin C = AB / AC = (x − a) / (x + a) = (4a − a) / (4a + a)
⇒ sin C = 3/5