# How do you find the exact value of the third side given triangle DEF, d=3.25, e=5.62, mangleF=58?

Jan 13, 2017

#### Answer:

$f \approx 4.7739 \left(4 \mathrm{dp}\right)$.

#### Explanation:

We apply the the Cosine Rule to $\Delta D E F$ to get,

$\cos F = \frac{{d}^{2} + {e}^{2} - {f}^{2}}{2 \mathrm{de}}$, or,

${f}^{2} = {d}^{2} + {e}^{2} - 2 \mathrm{de} \cos F$

$= {3.25}^{2} + {5.62}^{2} - 2 \left(3.25\right) \left(5.62\right) \cos 58$

$\approx 10.5625 + 31.5844 - 36.53 \left(0.5299\right)$

$= 42.1469 - 19.3572$

$= 22.7897$

$\therefore f \approx \sqrt{22.7897} \approx 4.7739 \left(4 \mathrm{dp}\right)$.