Which equation can be solved to find one of the missing side lengths in a triangle using the $Law of Sines$?

• A. $a+b+c=180$
• B. $a:b:c=A:B:C$
• C. $a=bcosC+ccosB$
• D. $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Answer: (D)

Verified step by step guidance

1. Recognize that the Law of Sines relates the ratios of the lengths of sides of a triangle to the sines of their opposite angles. The formula is given by: \(\frac{a}{\sin\left(A\right)} = \frac{b}{\sin\left(B\right)} = \frac{c}{\sin\left(C\right)}\).
2. Identify the known sides and angles in the triangle. To find a missing side length, you need at least one side length and its opposite angle, plus another angle or side.
3. Set up the equation using the Law of Sines by pairing the known side and angle ratio with the unknown side and its opposite angle. For example, if you know side \(b\) and angle \(B\), and want to find side \(a\), use \(\frac{a}{\sin\left(A\right)} = \frac{b}{\sin\left(B\right)}\).
4. Rearrange the equation to solve for the missing side length. For example, multiply both sides by \(\sin\left(A\right)\) to isolate \(a\): \(a = b \times \frac{\sin\left(A\right)}{\sin\left(B\right)}\).
5. Use the known angle measures and side lengths to substitute into the equation and solve for the missing side length.