Question

If the measure of angle A = 55 degrees, b = 12, and c = 7, then find the measure of angle C.

Please help I tried everything​

Answer 1

Check the picture below.

so firstly let's find side "a"

`\textit{Law of Cosines}\\\\ c^2 = a^2+b^2-(2ab)\cos(C)\implies c = √(a^2+b^2-(2ab)\cos(C)) \\\\[-0.35em] ~\dotfill\\\\ a = √(7^2+12^2~-~2(7)(12)\cos(55^o)) \implies a = √( 193 - 14112 \cos(55^o) ) \\\\\\ a \approx √( 193 - (96.3608) ) \implies a \approx √( 96.6392 ) \implies a \approx 9.8305`

now, let's use "a" to get angle C

`\textit{Law of Sines} \\\\ \cfrac{\sin(\measuredangle A)}{a}=\cfrac{\sin(\measuredangle B)}{b}=\cfrac{\sin(\measuredangle C)}{c} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{\sin( C )}{7}\approx\cfrac{\sin( 55^o )}{9.8305}\implies 9.8305\sin(C)\approx7\sin(55^o) \implies \sin(C)\approx\cfrac{7\sin(55^o)}{9.8305} \\\\\\ C\approx\sin^(-1)\left( ~~ \cfrac{7\sin( 55^o)}{9.8305} ~~\right)\implies \boxed{C\approx 35.7^o}`

Make sure your calculator is in Degree mode.