# A triangle has sides A, B, and C. Sides A and B are of lengths 1 and 7, respectively, and the angle between A and B is (5pi)/6 . What is the length of side C?

Feb 20, 2017

You can utilize the law of cosines to solve this problem.

#### Explanation:

Given some triangle with two sides and their included angle defined, we can solve for the third side. The notation I am following is that where $a , b , c$ denote the sides, while $\angle A , \angle B , \angle C$ denote the angles opposite those sides. The formula is given as follows:

${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos \left(\angle C\right)$

One can consider this the extended or general form of the pythagorean theorem, in which we have a compensation term to account for non-right triangles. To study this a bit, consider a focus on the angle, C. Should it be greater than 90 degrees, the cosine is negative, meaning there is an extra contribution, and c must be longer. This is true if you imagine the side opposite of an obtuse angle. Imagine C as 90, which reduces contribution to 0. Thus we have a normal right triangle and we are simply finding the hypotenuse. Given an acute angle, the third side must necessarily be smaller.

So how do we use this formula? Plug in values, and since we are solving for c, square root at the end.

We have the following:

$a = 1$

$b = 7$

$\angle C = \frac{5 \pi}{6}$

Simply plug them in, and evaluate, and this should yield a final answer that is close to ~8. Try it out!