Identify the missing side between two sides and an angle in a triangle

Expert intro: The wording here is a little messy, but the idea looks like a triangle problem where two side lengths and an angle are given, and you need the side that sits between them or the angle between them. In these cases, the first question is always: is the angle included between the known sides?

Detailed walkthrough

If you know two sides and the included angle, the natural tool is the Law of Cosines:

$$c^2=a^2+b^2-2ab\cos C$$

Here, the key phrase is “between”. That usually means the angle is sandwiched by the two known sides.

How to read the situation

• If $66$ and $58$ are the two sides,
• and the angle between them is the one labeled $61^\circ$ or $89^\circ$,
• then that angle is the included angle.

What to do next

1. Match the angle to the two sides touching it.
2. Plug the two side lengths and included angle into the Law of Cosines.
3. Solve for the missing side.

For example, if $66$ and $58$ are adjacent to a $61^\circ$ angle, then the opposite side is

$$c=\sqrt{66^2+58^2-2(66)(58)\cos 61^\circ}$$

That is the structure you want.

Short answer

The missing side is found with the Law of Cosines, not with a simple side-to-angle ratio.

💡 Pitfall guide

The main mistake is picking the wrong angle just because it looks closest. For Law of Cosines, the angle must be the one between the two known sides. If you use a non-included angle, the whole setup breaks.

🔄 Real-world variant

If instead you were given two angles and one side, you would switch to the Law of Sines. If you were given all three sides, then you would use the Law of Cosines again but in a different form to find an angle.

🔍 Related terms

included angle, Law of Cosines, triangle sides