A triangle has sides A, B, and C. Sides A and B are of lengths 6 and 8, respectively, and the angle between A and B is (5pi)/8 . What is the length of side C?

Apr 15, 2016

≈ 11.69

Explanation:

Given a triangle with 2 sides and the angle between them known.
Then to find the third side we use the$\textcolor{b l u e}{\text{ cosine rule }}$

color(red)(|bar(ul(color(white)(a/a)color(black)( c^2 = a^2 + b^2 - (2ab costheta)color(white)(a/a)|)))
where a , b are the 2 known sides , $\theta \text{ is the angle between them and c is the side to be found }$

here a = 6 , b = 8 and $\theta = \frac{5 \pi}{8}$

substitute these values into the formula.

${c}^{2} = {6}^{2} + {8}^{2} - \left(2 \times 6 \times 8 \times \cos \left(\frac{5 \pi}{8}\right)\right)$

= 36 + 64 - ( -36.738) = 136.738

now c^2 = 136.738 rArr c = sqrt136.738 ≈ 11.69