Question

solve the right triangle. round your answers to the nearest tenth. b = \\(\square^\circ\\) a = \\(\square\\) c = \\(\square\\) (image of a right triangle with right angle, side 30, angle 55° at the right angles adjacent vertex, side a opposite angle b, side c as the hypotenuse)

Explanation:

Step1: Find angle B

In a right triangle, the sum of angles is \(180^\circ\), and one angle is \(90^\circ\), another is \(55^\circ\). So \(B = 180^\circ - 90^\circ - 55^\circ = 35^\circ\).

Step2: Find side a (opposite to \(55^\circ\), adjacent is 30)

Using tangent: \(\tan(55^\circ)=\frac{a}{30}\), so \(a = 30\times\tan(55^\circ)\). \(\tan(55^\circ)\approx1.428\), so \(a\approx30\times1.428 = 42.8\) (rounded to nearest tenth).

Step3: Find side c (hypotenuse, adjacent to \(55^\circ\) is 30)

Using cosine: \(\cos(55^\circ)=\frac{30}{c}\), so \(c=\frac{30}{\cos(55^\circ)}\). \(\cos(55^\circ)\approx0.574\), so \(c\approx\frac{30}{0.574}\approx52.3\) (rounded to nearest tenth).

Answer:

\(B = 35^\circ\), \(a \approx 42.8\), \(c \approx 52.3\)