Question

1. solve the triangle:

(image of triangle with vertices a, b, c; angle at a is 132°, angle at b is 28°, side c (ab) = 6.8; labels for sides a (bc), b (ac), and angles a, b, c with blanks to fill: a =, a =; b =, b =; c =, c =)

Explanation:

Step1: Calculate angle C

Sum of angles in triangle is $180^\circ$.
$C = 180^\circ - 132^\circ - 28^\circ = 20^\circ$

Step2: Find side a with Law of Sines

$\frac{a}{\sin B} = \frac{c}{\sin C}$
$a = \frac{c \cdot \sin B}{\sin C} = \frac{6.8 \cdot \sin 28^\circ}{\sin 20^\circ}$
$\sin 28^\circ \approx 0.4695$, $\sin 20^\circ \approx 0.3420$
$a \approx \frac{6.8 \cdot 0.4695}{0.3420} \approx 9.3$

Step3: Find side b with Law of Sines

$\frac{b}{\sin A} = \frac{c}{\sin C}$
$b = \frac{c \cdot \sin A}{\sin C} = \frac{6.8 \cdot \sin 132^\circ}{\sin 20^\circ}$
$\sin 132^\circ = \sin(48^\circ) \approx 0.7431$
$b \approx \frac{6.8 \cdot 0.7431}{0.3420} \approx 14.8$

Answer:

$a \approx 9.3$, $b \approx 14.8$, $c = 6.8$, $A = 132^\circ$, $B = 28^\circ$, $C = 20^\circ$