Which rule could be used to find the length of an unknown side of a triangle given the measures of two angles and the length of one other side?
Let’s begin by sketching a triangle 𝐴𝐵𝐶 as shown. We will let the side length opposite angle 𝐴 be lowercase 𝑎. Likewise, we will let the side length opposite angle 𝐵 be lowercase 𝑏, and finally the side length opposite angle 𝐶 be lowercase 𝑐.
In this question, we need to determine which rule can be used when we are given the measures of two angles and the length of one side and need to calculate the length of unknown side. Let’s assume that we were given the measures of angles 𝐴 and 𝐵 and that we were also given the length of side 𝐴𝐶, which we have labeled lowercase 𝑏. Let’s also assume that we are trying to find the length of side 𝐵𝐶, which we have labeled lowercase 𝑎.
As there will be two angles and two side lengths involved in our calculation, we can use the sine rule or law of sines. This states that 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵 which is equal to 𝑐 over sin 𝐶, where lowercase 𝑎, 𝑏, and 𝑐 are the lengths of the sides of the triangle and capital 𝐴, 𝐵, and 𝐶 are the measures of the angles opposite these sides.
Specifically, in our example, we will use the fact that 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵. After multiplying through by sin 𝐴, we can find the length of the unknown side by performing the calculation 𝑏 over sin 𝐵 multiplied by sin 𝐴. We can find the length of an unknown side of a triangle given the measures of two angles and the length of one other side using the sin rule.
It is important to note that it doesn’t matter which two angles we are given. We know that the three angles in a triangle sum to 180 degrees. This means that if we are given any two angles, we can calculate the third one if this is required in the sine rule. Likewise, it doesn’t matter which side length we are given, as if we know the measures of two angles and the length of one side in a triangle, we will always be able to calculate the length of an unknown side using the sine rule.