Question

In the given problem solve the triangle using the law of sines :$A=50°,C=20°,a=3$.

Short answer

Required values of the triangle are$B=110°,c=1.3,b=3.7$.

Step-by-step solution

Step 1. Given information

For a triangle with sides a, b, c and opposite angles A, B, C, respectively, Law of sines is given as:

$\raisebox{1ex}{$\sin \left(A\right)$}\!\left/ \!\raisebox{-1ex}{$a$}\right.=\raisebox{1ex}{$\sin \left(B\right)$}\!\left/ \!\raisebox{-1ex}{$b$}\right.=\raisebox{1ex}{$\sin \left(C\right)$}\!\left/ \!\raisebox{-1ex}{$c$}\right.$

Step 2. Calculation

$$\begin{gathered} Given, \\ A=50°,C=20°,a=3 \\ Forc, \\ AccordingtoLawof\sin es \\ \Rightarrow \raisebox{1ex}{$\sin \left(A\right)$}\!\left/ \!\raisebox{-1ex}{$a$}\right.=\raisebox{1ex}{$\sin \left(C\right)$}\!\left/ \!\raisebox{-1ex}{$c$}\right. \\ \Rightarrow c=\raisebox{1ex}{$\sin \left(C\right)\times a$}\!\left/ \!\raisebox{-1ex}{$\sin \left(A\right)$}\right. \\ =\raisebox{1ex}{$\sin \left(20°\right)\times 3$}\!\left/ \!\raisebox{-1ex}{$\sin \left(50°\right)$}\right. \\ =\raisebox{1ex}{$0.342\times 3$}\!\left/ \!\raisebox{-1ex}{$0.766$}\right. \\ =\raisebox{1ex}{$1.026$}\!\left/ \!\raisebox{-1ex}{$0.766$}\right. \\ =1.339\approx 1.3 \\ ForB, \\ \Rightarrow B=180°-A-C \\ =180°-50°-20° \\ =180°-70° \\ =110° \\ Forb, \\ AccordingtoLawof\sin es \\ \Rightarrow \raisebox{1ex}{$\sin \left(A\right)$}\!\left/ \!\raisebox{-1ex}{$a$}\right.=\raisebox{1ex}{$\sin \left(B\right)$}\!\left/ \!\raisebox{-1ex}{$b$}\right. \\ \Rightarrow b=\raisebox{1ex}{$\sin \left(B\right)\times a$}\!\left/ \!\raisebox{-1ex}{$\sin \left(A\right)$}\right. \\ =\raisebox{1ex}{$\sin \left(110°\right)\times 3$}\!\left/ \!\raisebox{-1ex}{$\sin \left(50°\right)$}\right. \\ =\raisebox{1ex}{$0.9396\times 3$}\!\left/ \!\raisebox{-1ex}{$0.766$}\right. \\ =3.679\approx 3.7 \end{gathered}$$