Question

solve the following right triangle. determine all the missing sides and angles. a. $overline{bc}=5cm$ $overline{ac}=4cm$ $angle b = 27^{circ}$ $angle c=90^{circ}$

Explanation:

Step1: Find angle B

The sum of angles in a triangle is 180°. In a right - triangle $\angle C = 90^{\circ}$ and $\angle A=53^{\circ}$. So, $\angle B=180^{\circ}-\angle A - \angle C$.
$\angle B=180^{\circ}-53^{\circ}-90^{\circ}=37^{\circ}$

Step2: Find side BC

We know $\sin A=\frac{BC}{AB}$. Given $AB = 6.0$ cm and $\angle A = 53^{\circ}$, and $\sin53^{\circ}\approx0.7986$. Then $BC = AB\times\sin A$.
$BC=6.0\times\sin53^{\circ}\approx6.0\times0.7986 = 4.7916\approx4.8$ cm

Step3: Find side AC

We know $\cos A=\frac{AC}{AB}$. Given $AB = 6.0$ cm and $\angle A = 53^{\circ}$, and $\cos53^{\circ}\approx0.6018$. Then $AC = AB\times\cos A$.
$AC=6.0\times\cos53^{\circ}\approx6.0\times0.6018 = 3.6108\approx3.6$ cm

Answer:

$\overline{BC}\approx4.8$ cm, $\overline{AC}\approx3.6$ cm, $\angle B = 37^{\circ}$