Question

In a triangle ABC, with usual notations, if $a = 5$, $b = 7$, $\sin A = \frac{3}{4}$, then total number of triangles possible are ______.

• A. 1
• B. 0
• C. 2
• D. 5

Hint:

Always check the output of the Law of Sines immediately! If you get $\sin \theta > 1$, stop calculating. The triangle is physically impossible (0 triangles).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This problem tests the "Ambiguous Case" of solving triangles (SSA condition). We are given two side lengths and a non-included angle. We must determine how many physically valid triangles can be constructed.

Step 2: Detailed Explanation:
We are given:
$a = 5$
$b = 7$
$\sin A = \frac{3}{4}$
To see if a triangle is possible, apply the Law of Sines:
$\frac{a}{\sin A} = \frac{b}{\sin B}$
Rearrange to solve for $\sin B$, because angle $B$ determines the validity of the geometry:
$\sin B = \frac{b \cdot \sin A}{a}$
Substitute the known values into the equation:
$\sin B = \frac{7 \cdot (3/4)}{5}$
$\sin B = \frac{21/4}{5}$
$\sin B = \frac{21}{20}$
Evaluate the resulting value:
$\sin B = 1.05$
A fundamental property of real geometry and trigonometry is that the sine of any real angle must strictly lie within the closed interval $[-1, 1]$.
Because the calculation requires $\sin B$ to be greater than 1 ($1.05 > 1$), no such real angle $B$ can mathematically or physically exist.
Therefore, it is completely impossible to construct any triangle with these specific dimensions.

Step 3: Final Answer:
The total number of triangles possible is 0, matching option (b).