Question

In ΔABC, if ∠A = 45°, ∠B = 60° then find the ratio of its sides.

Answer 1

By the sine rule,
`a/"sin A" = b/"sin B" = c/"sin C"`

∴ `a/b = "sin A"/"sin B" and b/c = "sin B"/"sin C"`

∴ a : b : c = sinA : sinB : sinC

Given ∠A = 45° and ∠B = 60°
∵ ∠A + ∠B + ∠C = 180°
∴ 45° + 60° + ∠C = 180°
∴ ∠C = 180° – 105° = 75°

Now, sin A = sin 45° = `(1)/(√2)`

sin B = sin 60° = `(√3)/(2)`

and  sin C = sin 75° = sin( 45° + 30°)
= sin 45° cos 30° + cos 45° sin 30°

= `(1)/(√2) xx (√3)/(2) + (1)/(√2) xx (1)/(2)`

= `(√3)/(2(√2)) + (1)/(2(√2))`

= `(√3 + 1)/(2(√2))`
∴ the ratio of the sides of ΔABC
= a : b : c
= sinA : sinB : sinC
= `(1)/(√2) : (√3)/(2) : (√3 + 1)/(2√2)`
∴ a : b : c = 2: √6: (√3 + 1).