Question

1. list two cases in which we would use the cosine law (include diagrams).

case 1:
case 2:

Explanation:

The problem is about the cosine law in Mathematics (subfield: Geometry). The cosine law relates the lengths of the sides of a triangle to the cosine of one of its angles.

• Case 1: When we know two sides and the included angle of a triangle (SAS - Side - Angle - Side) and need to find the third side. For example, in triangle \(ABC\), if we know \(AB = c\), \(AC = b\), and \(\angle A=\alpha\), we can find \(BC = a\) using \(a^{2}=b^{2}+c^{2}-2bc\cos\alpha\). The diagram would be a triangle with two sides and the included angle labeled.
• Case 2: When we know all three sides of a triangle (SSS - Side - Side - Side) and need to find any of the angles. For example, in triangle \(ABC\) with sides \(a\), \(b\), \(c\), we can find \(\angle A\) using \(\cos\alpha=\frac{b^{2}+c^{2}-a^{2}}{2bc}\). The diagram would be a triangle with all three sides labeled.

Answer:

• Case 1: Given two sides and the included angle (SAS) of a triangle, to find the third side. Diagram: A triangle with two sides (e.g., \(AB\), \(AC\)) and the included angle (\(\angle A\)) labeled, and the third side (\(BC\)) to be found. Formula: \(a^{2}=b^{2}+c^{2}-2bc\cos\alpha\) (where \(a = BC\), \(b = AC\), \(c = AB\), \(\alpha=\angle A\)).
• Case 2: Given all three sides (SSS) of a triangle, to find an angle. Diagram: A triangle with all three sides (\(a\), \(b\), \(c\)) labeled, and an angle (e.g., \(\angle A\)) to be found. Formula: \(\cos\alpha=\frac{b^{2}+c^{2}-a^{2}}{2bc}\) (where \(a\), \(b\), \(c\) are the sides, \(\alpha=\angle A\)).