Question

In the triangle \(ABC\), if \(a=7\), \(b=6\) and \(A=120^\circ\), then the approximate value of \(B\) is

• A. \(47.9^\circ\)
• B. \(44.9^\circ\)
• C. \(59.9^\circ\)
• D. \(61.9^\circ\)

Hint:

When one angle of a triangle is obtuse, the other two angles must be acute. So while using the sine rule, reject any angle value that makes the angle sum exceed \(180^\circ\).

Answer 1

The Correct Option is A

Step 1: Use the sine rule.
In any triangle, \[ \frac{a}{\sin A}=\frac{b}{\sin B} \] Therefore, \[ \frac{7}{\sin 120^\circ}=\frac{6}{\sin B} \]

Step 2: Find \(\sin B\).
From the sine rule, \[ \sin B=\frac{6\sin 120^\circ}{7} \] We know that \[ \sin 120^\circ=\frac{\sqrt3}{2} \] So, \[ \sin B=\frac{6}{7}\times \frac{\sqrt3}{2} \] \[ \sin B=\frac{3\sqrt3}{7} \] \[ \sin B\approx 0.742 \]

Step 3: Find the angle \(B\).
Thus, \[ B=\sin^{-1}(0.742) \] \[ B\approx 47.9^\circ \] Since \[ A=120^\circ, \] the remaining two angles together must be \[ 60^\circ \] Therefore, \(B\) must be less than \(60^\circ\), so the valid value is \[ 47.9^\circ \]

Step 4: Final conclusion.
Therefore, \[ \boxed{47.9^\circ} \]