# How do you solve a^2=3^2+5^2-2(3)(5)cos85?

May 1, 2018

$a = 31.4$ units

#### Explanation:

${a}^{2} = {3}^{2} + {5}^{2} - \left(2 \cdot 3 \cdot 5\right) \cos {85}^{\circ}$.

it seems best to first simplify the expression on the right-hand side.

${3}^{2} + {5}^{2} = 9 + 25 = 34$

$2 \cdot 3 \cdot 5 = 30 \rightarrow - \left(2 \cdot 3 \cdot 5\right) \cos {85}^{\circ} = - 30 \cos {85}^{\circ}$/

${3}^{2} + {5}^{2} - \left(2 \cdot 3 \cdot 5\right) \cos {85}^{\circ} = 34 - 30 \cos {85}^{\circ}$.

hence, ${a}^{2} = 34 - 30 \cos {85}^{\circ}$.

since side length cannot be negative, $a$ is the positive square root of $34 - 30 \cos {85}^{\circ}$.

putting $\sqrt{34 - 30 \cos {85}^{\circ}}$ into a calculator gives $31.38532771756$.

this can be rounded to $3$ significant figures to give $31.4$.

therefore $a = 31.4$ units in length.