Question

instructions: round trigonometric rati round length answers to 1. solve the following triangles. a) a 55° b c c 60 32° b 77 cm 9

Explanation:

Step1: Find angle A

The sum of angles in a triangle is 180°. So, $A=180^{\circ}-(60^{\circ}+32^{\circ}) = 88^{\circ}$

Step2: Use the Law of Sines to find side b

The Law of Sines states $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. We know $a = 77$ cm, $A=88^{\circ}$, $B = 32^{\circ}$. So, $\frac{b}{\sin B}=\frac{a}{\sin A}$, then $b=\frac{a\sin B}{\sin A}=\frac{77\sin32^{\circ}}{\sin88^{\circ}}$.
$\sin32^{\circ}\approx0.5299$, $\sin88^{\circ}\approx0.9994$. So, $b=\frac{77\times0.5299}{0.9994}\approx40.9$ cm.

Step3: Use the Law of Sines to find side c

Using $\frac{c}{\sin C}=\frac{a}{\sin A}$, with $C = 60^{\circ}$, $a = 77$ cm, $A=88^{\circ}$. Then $c=\frac{a\sin C}{\sin A}=\frac{77\sin60^{\circ}}{\sin88^{\circ}}$.
$\sin60^{\circ}=\frac{\sqrt{3}}{2}\approx0.8660$, $\sin88^{\circ}\approx0.9994$. So, $c=\frac{77\times0.8660}{0.9994}\approx66.9$ cm.

Answer:

$A = 88^{\circ}$, $b\approx40.9$ cm, $c\approx66.9$ cm