Given triangle ABC with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$ respectively, which of the following correctly expresses the Law of Sines?

• A. $a+b+c=180$
• B. $\frac{a}{b}=\frac{c}{A}$
• C. $\frac{a}{cos\left(A\right)}=\frac{b}{cos\left(B\right)}=\frac{c}{cos\left(C\right)}$
• D. $\frac{a}{sin\left(A\right)}=\frac{b}{sin\left(B\right)}=\frac{c}{sin\left(C\right)}$

Answer: (D)

Verified step by step guidance

1. Recall that the Law of Sines relates the sides of a triangle to the sines of their opposite angles. It is a fundamental rule in trigonometry for any triangle, not just right triangles.
2. Identify the sides and angles in the triangle: side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\), and side \(c\) is opposite angle \(C\).
3. Write the Law of Sines formula, which states that the ratio of a side length to the sine of its opposite angle is constant for all three sides and angles in the triangle:
4. \[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\]
5. Understand that this formula allows you to find unknown sides or angles in a triangle when you know some combination of sides and angles, making it very useful in solving triangle problems.