Krishna
0

Step 1:  Recall the law of sines formula

              FORMULA: \frac{a}{\sin A} = \frac{b}{ \sin B} = \frac{c}{\sin C}

​              Here each lowercase letter (like a) is the length of the side

              opposite the vertex labeled with the same capital letter (like A).


Step 2: Find the unknown angle by using the law of sines formula  

            GIVEN: In a triangle PQR

                                 \angle P = 116\degree

                                            p  = 8.3 cm

                                            q  = 5.4 cm

                                              r = ?

                    Sine formula;

                                 \frac{p}{\sin P} = \frac{q}{\sin Q}


                               \frac{8.3}{\sin 116\degree} = \frac{5.4}{\sin Q}


                               \sin Q\degree = \frac{5.4*\sin 116\degree}{8.3}


                                 \sin Q = \frac{5.4 * 0.8987}{8.3}


                             Q = \sin^{-1}(0.5848)


                             Q = 35.8\degree or 144.2\degree


                  Q cannot be an obtuse angle because the sum of angles in the triangle

                  will exceed 180\degree. The only valid value for

                   Q is 35.8\degree.


              Find the remaining angle in the triangle

                         \angle P + \angle Q + \angle R = 180\degree

                       116\degree + \angle 35.8\degree + \angle R = 180\degree

                             \angle R = 180 - 35.8 = 28.2\degree


Step 3: Calculate the unknown length by using the law of sines


             FORMULA:       \frac{8.3}{\sin 116\degree} = \frac{r}{\sin 28\degree} 


                                         \frac{8.3 * \sin 28\degree}{\sin 116\degree} = r  


                                           r = \frac{(8.3 * 4.69) }{0.8987}


                                              r = 4.36