Question

Which of the following pieces of data does not uniquely determine acute angled triangle \[ABC\left( R=circum-radius \right)\]
(A) \[a,\sin A,\sin B\]
(B) \[a,b,c\]
(C) \[a,\sin B,R\]
(D) \[a,\sin A,R\]

Answer 1

Hint: We are given a question asking to find the options which when employed does not give a unique value in order to determine whether the triangle is an acute angled triangle \[ABC\left( R=circum-radius \right)\]. Here, we will make use of the Law of Sines, which is represented as, \[\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R\]. We will take up each of the options and determine if they can uniquely obtain the values of the angles of the given triangle. Hence, we will have the correct option.

Complete step by step answer:
According to the given question, we are given a question asking us to choose a particular option which does not uniquely determine the acute angled triangle \[ABC\left( R=circum-radius \right)\].
We will make use of the expression of Law of Sines, which is, \[\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R\]
We will take each of the options one by one and will determine the required values.
The law of sines expression that we have is,
\[\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R\]
We can also write, \[C=\pi -\left( A+B \right)\], that is, we get,
\[\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin \left( A+B \right)}=2R\]
(A) \[a,\sin A,\sin B\]
Making use of the above expression of Law of sines, we can find the values of A, B and C. That is, we can uniquely determine the values of the angles of the given triangle \[ABC\]. So, this is an incorrect option.
Next, we have,
(B) \[a,b,c\]
If we are given the sides, we can very well determine the angles associated with the triangle using the ‘Law of Cosines’. So, this option is an incorrect option.
Now, we have,
(C) \[a,\sin B,R\]
Here, with the side ‘a’ and R (radius), we can determine the angle A. Using the modification of the previous step we can find the value of the angle B and since \[C=\pi -\left( A+B \right)\]. So, we can uniquely obtain each of the angles and so this option is incorrect.
Next, we got,
(D) \[a,\sin A,R\]
Here, we are only given ‘a’ and the sine of the angle A and of course ‘R’. Since we do not have the ‘b’ values so we cannot determine the angle B and hence the value of C can also not be determined. That is, we have,
\[\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R\]

So, the correct answer is “Option D”.

Note: The Law of sines must be known and its expression should be correctly written. We have to determine the values of A, B and C mainly, so while checking each of the options, make sure every way is considered to find the values using the given piece of data in each of the options. And acute angled triangles have angles less than \[{{90}^{\circ }}\].