Which of the following pieces of data does NOT uniquely determine an acute-angled triangle `A B C(R` being the radius of the circumcircle)? (a) `a ,sinA ,sinB` (b) `a , b , c (c) `a ,sinB ,R` (d) `a ,sinA ,R`

Options

A. `a, sin A, sin B`
B. `a, b, c`
C. `a, sin B, R`
D. `a, sin A, R`

Explanation

By sine law in `Delta ABC`, we have
`(a)/(sinA) = (b)/(sin B) = (c)/(sin (pi - A - B)) = 2R`
or `(a)/(sinA) = (b)/(sin B) = (c)/(sin (A + B)) = 2R`
(1) If we know a, sin A, sin B, we can find b, c, and the value of angle A,B, C
(2) With a, b, c we can find `angleA , angle B, angle C` using the cosine law.
(3) a, sin B, R are given, so sin A, b and hence `sin (A + B)` and then C be found
(4) If we know a, sin A , R, then we can get the ratio `(b)/(sin B) " or " (c)/(sin (A + B))` only. We cannot determine the values of b, c, sin B, sin C separately. Therefore, the triangle cannot be determined uniquely in this case