In which of the following cases, there exists a triangle ABC?
(a) `b sin A = a, A lt pi//2`
(b) `b sin A gt a, A gt pi//2`
(c) `b sin A gt a, A lt pi//2`
(d) `b sin A lt a, A lt pi//2, b gt a`
(e) `b sin A lt a, A gt pi//2, b = a`

Answer

(a),(b)

Explanation

We have `(sin A)/(a) = (sin B)/(b)`
`rArr a sin B = b sin A`
(a) `b sin A = a rArr a sin B = a`
since `A lt pi//2, " the " Delta ABC` is possible.
(b) `b sin A gt a rArr a sin B gt a rArr sin B gt 1`
Which is impossible. Hence, the possibiltity (ii) is rule out
`b sin A gt a, A lt pi//2`
`rArr a sin B gt a`
`rArr sin B gt 1` (which is not possible)
(d) `b sin A lt a rArr a sin B lt a rArr sin B lt 1`
So, value of `angle B` exists.
Now, `b gt a rArr B gt A`. Since `A lt pi//2`
The `Delta ABC` is possible when `B gt pi//2`
(e) Since `b = a`, we have `B = A`, But `A gt pi//2`
Therefore, `B gt pi//2`. But this is not possible for any triangle