In triangle $DEF$, side $d$ = $8$, side $e$ = $12$, and angle $D$ = $40°$. Using the Law of Sines, what is the approximate measure of angle $F$?

• A. $48°$
• B. $72°$
• C. $40°$
• D. $60°$

Answer: (B)

Verified step by step guidance

1. Identify the given elements in triangle DEF: side d = 8, side e = 12, and angle D = 40°.
2. Recall the Law of Sines formula: \(\frac{d}{\sin(D)} = \frac{e}{\sin(E)} = \frac{f}{\sin(F)}\), where d, e, f are sides opposite angles D, E, F respectively.
3. Set up the equation relating sides d and e with their opposite angles D and E: \(\frac{8}{\sin(40^\circ)} = \frac{12}{\sin(E)}\).
4. Solve for \(\sin(E)\) by cross-multiplying: \(\sin(E) = \frac{12 \times \sin(40^\circ)}{8}\).
5. Calculate \(\sin(E)\) using the known values, then find angle E by taking the inverse sine (arcsin) of that value. Finally, find angle F by using the triangle angle sum property: \(F = 180^\circ - D - E\).