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 Concept:
Use the Sine Rule $\frac{a}{\sin A} = \frac{b}{\sin B}$ to check the validity of $\sin B$. For a triangle to exist, $\sin B$ must be $\le 1$.

Step 2: Formula Application:
$\sin B = \frac{b \sin A}{a} = \frac{7 \times (3/4)}{5} = \frac{21}{20}$.

Step 3: Explanation:
$\sin B = 1.05$. Since the value of the sine function cannot exceed 1, no such angle $B$ exists. Therefore, no triangle can be formed with these specific dimensions.

Step 4: Final Answer:
The total number of triangles possible is 0.