Question

use the triangle shown on the right and the given information to solve the triangle
$b = 6$, $a = 70^{circ}$, find $a$, $c$, and $b$
$a=square$ (round to two decimal places as needed )

Explanation:

Step1: Find angle \( B \)

In a right - triangle, \( A + B=90^{\circ} \). Given \( A = 70^{\circ} \), then \( B=90^{\circ}-A \).
\( B = 90^{\circ}-70^{\circ}=20^{\circ} \)

Step2: Find side \( a \)

We know that \( \tan A=\frac{a}{b} \). Given \( b = 6 \) and \( A = 70^{\circ} \), then \( a=b\tan A \).
Since \( \tan70^{\circ}\approx2.747 \), \( a = 6\times\tan70^{\circ}\approx6\times2.747 = 16.48 \)

Step3: Find side \( c \)

We know that \( \cos A=\frac{b}{c} \), then \( c=\frac{b}{\cos A} \). Given \( b = 6 \) and \( A = 70^{\circ} \), and \( \cos70^{\circ}\approx0.342 \)
\( c=\frac{6}{\cos70^{\circ}}\approx\frac{6}{0.342}\approx17.54 \)

Answer:

\( a\approx16.48 \), \( c\approx17.54 \), \( B = 20^{\circ} \)