How do you solve the triangle RST given R=35^circ, s=16, t=9?

May 20, 2017

You're given the included angle. Use the Law of Cosines to solve this problem.
The formula : ${a}^{2} = {b}^{2} + {c}^{2} - 2 b c \cdot \cos A$
Let's replace $a$ with $r$, $b$ with $s$, and $c$ with $t$.
Let's solve for side $r$ first.
Substitute values.
${r}^{2} = {16}^{2} + {9}^{2} - \left(2 \cdot 16 \cdot 9\right) \cdot \cos \left(35\right)$
${r}^{2} = 256 + 81 - 288 \cdot \cos \left(35\right)$
${r}^{2} \approx 337 - 235.92$
${r}^{2} \approx 101.09$
$r \approx 10.05$

Now, let's solve for angle S.
You can solve using the Law of Sines, but I'll just use the Law of Cosines because you didn't seem to know how to do it.

Also, for the sake of simplicity, I'll just use $r = 10$
The rearranged formula of the Law of Cosines (and changed to our variables r, s and t) : $\cos S = \frac{{r}^{2} + {t}^{2} - {s}^{2}}{2 r t}$
Substitute values :
$\cos S = \frac{{10}^{2} + {9}^{2} - {16}^{2}}{2 \cdot 16 \cdot 9}$
$\cos S = \frac{100 + 81 - 256}{288}$
$\cos S = - \frac{25}{96}$
$S = {\cos}^{-} 1 \left(- \frac{25}{96}\right)$
$S \approx 74.91$

You can work out the remaining angle using the angles in a triangle add up to 180 rule.