We have two sides of the triangle (a=8 and b=16) and an angle (30°).
We will assume that the known angle is between the known sides a and b.
We can start by calculating the third side "c" with the Law of cosines:
[tex]\begin{gathered} c^2=a^2+b^2-2ab\cdot\cos (A) \\ c^2=8^2+16^2-2\cdot8\cdot16\cdot\cos (30\degree) \\ c^2=64+256-256\cdot\frac{\sqrt[]{3}}{2} \\ c^2=320-128\sqrt[]{3} \\ c^2\approx320-221.7 \\ c\approx\sqrt[]{98.3} \\ c\approx9.9 \end{gathered}[/tex]Now, we can use the Law of sines, where the quotient between the sine of an angle and its opposite side is constant for each of the angles:
[tex]\frac{\sin(A)}{c}=\frac{\sin (B)}{a}=\frac{\sin(C)}{b}[/tex]We already know the value of the first term:
[tex]\frac{\sin(A)}{c}=\frac{\sin (30\degree)}{9.9}=\frac{0.5}{9.9}=\frac{5}{99}[/tex]Then we can calculate the measure of B as:
[tex]\begin{gathered} \sin (B)=a\cdot\frac{\sin(A)}{c}=8\cdot\frac{5}{99}\approx0.404 \\ B=\arcsin (0.404) \\ B\approx23.8\degree \end{gathered}[/tex]Now we can calculate the measure of C as:
[tex]\begin{gathered} A+B+C=180\degree \\ 30+23.8+C=180 \\ C=180-30-23.8 \\ C=126.2\degree \end{gathered}[/tex]Answer:
The sides of the triangle are: 8, 16 and 9.9.
The angles of the triangle are: 30°, 23.8° and 126.2°.
