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