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,sinA,sinB
B. a,b,c
C. a,sinB,R
D. a,sinA,R

Explanation

Using sine rule, we have
`a/(ainA)=b/(sinB)=c/(sin(pi-A-B))`
`rArra/(sinA)=b/(sinB)=c/(sin(A+B))`
This shows that we can determine b, c and C when we are given a, sin A and sin B.
When a, b, c are given, we can find A, B, C by using cosine formulae. By using
`a/(sinA)=b/(sinB)=c/(sinC)=2R`
we can find A, C, b and c, if we are given the values of a, sin B and R
We cannot find `angleB, angleC` and the sides b and c if we just know a, sin A and R, because
`a/(sinA)=b/(sinB)=c/(sinC)=2R`
gives us the ratio `b/(sinB)` and `c/(sinC)` from where one cannot obtain b,cB and C.