Question

solve the right triangle abc, with c = 90°. a = 36.8°, b = 34.5 cm b = □° (simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.) a ≈ □ cm (simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.) c ≈ □ cm (simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.)

Explanation:

Step1: Find angle B

In a right triangle, the sum of angles is \(180^\circ\), and \(C = 90^\circ\), \(A = 36.8^\circ\). So \(B=180^\circ - 90^\circ - 36.8^\circ\)
\(B = 53.2^\circ\)

Step2: Find side a (opposite to angle A, adjacent to angle B)

We know that \(\tan A=\frac{a}{b}\), so \(a = b\times\tan A\). Given \(b = 34.5\) cm, \(A = 36.8^\circ\), \(\tan(36.8^\circ)\approx0.75\) (since \(\tan(36.8^\circ)\approx\frac{3}{4}=0.75\) approximately)
\(a = 34.5\times\tan(36.8^\circ)\approx34.5\times0.75 = 25.875\approx25.9\) cm

Step3: Find side c (hypotenuse)

We can use \(\cos A=\frac{b}{c}\), so \(c=\frac{b}{\cos A}\). \(\cos(36.8^\circ)\approx0.8\) (since \(\cos(36.8^\circ)\approx\frac{4}{5}=0.8\) approximately)
\(c=\frac{34.5}{\cos(36.8^\circ)}\approx\frac{34.5}{0.8}=43.125\approx43.1\) cm (or using Pythagorean theorem: \(c=\sqrt{a^{2}+b^{2}}=\sqrt{(25.875)^{2}+(34.5)^{2}}\approx\sqrt{669.5156 + 1190.25}=\sqrt{1859.7656}\approx43.1\) cm)

Answer:

\(B = 53.2^\circ\)
\(a\approx25.9\) cm
\(c\approx43.1\) cm