if tan A : tan B : tan C = 1:2:3, then sin A : sin B : sin C

We are given that:

tan A : tan B : tan C = 1 : 2 : 3

Step 1: Express tan values in terms of angles
Let
tan A = k
tan B = 2k
tan C = 3k

Step 2: Recall the identity relating sine and tangent
Using the identity:

sin θ = tan θ / √(1 + tan² θ)

Step 3: Find each sine value
sin A = k / √(1 + k²)
sin B = 2k / √(1 + (2k)²) = 2k / √(1 + 4k²)
sin C = 3k / √(1 + (3k)²) = 3k / √(1 + 9k²)

Step 4: Finding the ratio
To simplify the ratio, multiply each term by √(1 + k²), √(1 + 4k²), and √(1 + 9k²) respectively:

sin A : sin B : sin C
= k / √(1 + k²) : 2k / √(1 + 4k²) : 3k / √(1 + 9k²)
= 1 / √(1 + k²) : 2 / √(1 + 4k²) : 3 / √(1 + 9k²)

Step 5: Final Answer
The required ratio is:

1/√(1 + k²) : 2/√(1 + 4k²) : 3/√(1 + 9k²)

This is the simplest form unless specific values are provided for k.