A triangle has sides A, B, and C. Sides A and B are of lengths 1 and 2, respectively, and the angle between A and B is (pi)/8 . What is the length of side C?

Feb 26, 2018

Answer for the updated question given below.
$c \approx 1.14$

Explanation:

$\textcolor{red}{\triangle A B C}$ has sidescolor(red)(AB, BC,and CA. Sides $\textcolor{red}{a}$ and $\textcolor{red}{b}$ are of lengths 1 and 2 respectively, and the angle between $\textcolor{red}{a}$ and $\textcolor{red}{b}$ is $\frac{\pi}{8}$. What is the length of side $\textcolor{red}{c}$?
Where$\textcolor{red}{A B = c , B C = a \mathmr{and} C A = b .}$
Here, $a = 1 , b = 2 , C = \frac{\pi}{8}$
Using the Law of Cosines:
$\cos C = \frac{{a}^{2} + {b}^{2} - {c}^{2}}{2 a b} \Rightarrow {c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos C$
Hence,${c}^{2} = {\left(1\right)}^{2} + {\left(2\right)}^{2} - 2 \left(1\right) \left(2\right) \cos \left(\frac{\pi}{8}\right) \approx 1 + 4 - 4 \left(0.9239\right) \Rightarrow {c}^{2} \approx 5 - 3.6956 = 1.3044$$\textcolor{red}{\Rightarrow c \approx 1.14}$