Question

In a triangle \[ABC\], \[a = 5\], \[b = 7\] and \[\sin A = \dfrac{3}{4}\]. Then find how many such triangles are possible.
A. 1
B. 0
C. 2
D. Infinite

Answer 1

Hint:
 Here, length of two sides of a triangle and sin ratio of an angle \[A\] is given. We will use the law of sines and find the value of the other angle \[B\]. If the value of the ratio is greater than 1, then triangles do not form.



Formula Used:
Law of sines: In a triangle \[ABC\], \[\dfrac{{\sin A}}{a} = \dfrac{{\sin B}}{b} = \dfrac{{\sin C}}{c}\]


Complete step-by-step answer:
Given:
In\[ \triangle ABC \] , \[a = 5\], \[b = 7\] and \[\sin A = \dfrac{3}{4}\].

Let’s apply the law of sines for the measurement of the given triangle.
We get,
\[\dfrac{{\sin A}}{a} = \dfrac{{\sin B}}{b}\]
Substitute the values in the above equation.
\[\dfrac{{\dfrac{3}{4}}}{5} = \dfrac{{\sin B}}{7}\]
\[ \Rightarrow \dfrac{3}{{20}} = \dfrac{{\sin B}}{7}\]
\[ \Rightarrow \sin B = \dfrac{{21}}{{20}}\]
We know that the range of the sine function is \[\left[ { - 1,1} \right]\] .
Here, \[\sin B > 1\].
Which is not possible.
So, the given measurements do not determine any triangle.
Hence the correct option is B.

Additional information:
There are two laws on an oblique triangle: The first law is sine law and the second law is cosine law.
To apply sine law we need to know at least 2 sides of the triangle and the angle between them or 2 angles and one opposite side of one of the angles.
To apply cosine law, we need two sides and one angle.



Note:
The law of sine or the sine law or sine rule states that the ratio of the length of sides of a triangle to the sine of the opposite angle of a triangle.