Given the Law of Sines, could $a$ and $b$ be the side lengths of a triangle if $a=3$ and $b=7$ and the angle opposite $a$ is $80°$ and the angle opposite $b$ is $40°$?

• A. No, because the sum of the angles is not $180°$.
• B. Yes, because the Law of Sines is satisfied for these values.
• C. Yes, because any two positive numbers can be side lengths of a triangle.
• D. No, because the Law of Sines is not satisfied for these values.

Answer: (D)

Verified step by step guidance

1. Recall the Law of Sines formula: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), where \(a\), \(b\), and \(c\) are side lengths opposite angles \(A\), \(B\), and \(C\) respectively.
2. Identify the given values: \(a = 3\), \(b = 7\), \(A = 80^\circ\), and \(B = 40^\circ\).
3. Calculate the ratio \(\frac{a}{\sin A}\) by substituting the known values: \(\frac{3}{\sin 80^\circ}\).
4. Calculate the ratio \(\frac{b}{\sin B}\) by substituting the known values: \(\frac{7}{\sin 40^\circ}\).
5. Compare the two ratios \(\frac{3}{\sin 80^\circ}\) and \(\frac{7}{\sin 40^\circ}\). If they are equal (or very close), the Law of Sines is satisfied and these could be side lengths of a triangle; if not, then these values cannot form a triangle according to the Law of Sines.