Question

1. in a right triangle, one acute angle is $35^{circ}$ and the side adjacent to this angle is 8 cm. find the measures of the other acute angle and the lengths of the other two sides. (round to two decimal places).

Explanation:

Step1: Find the other acute angle

In a right - triangle, the sum of the two acute angles is \(90^{\circ}\). Let the other acute angle be \(\theta\).
If one acute angle is \(35^{\circ}\), then \(\theta=90^{\circ}- 35^{\circ}=55^{\circ}\)

Step2: Find the opposite side (\(a\))

We know that \(\tan\alpha=\frac{\text{opposite}}{\text{adjacent}}\), where \(\alpha = 35^{\circ}\) and adjacent \(b = 8\) cm.
\(\tan35^{\circ}=\frac{a}{8}\), so \(a = 8\times\tan35^{\circ}\)
Since \(\tan35^{\circ}\approx0.7002\), then \(a=8\times0.7002 = 5.60\) cm

Step3: Find the hypotenuse (\(c\))

We know that \(\cos\alpha=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\alpha = 35^{\circ}\), adjacent \(b = 8\) cm
\(\cos35^{\circ}=\frac{8}{c}\), so \(c=\frac{8}{\cos35^{\circ}}\)
Since \(\cos35^{\circ}\approx0.8192\), then \(c=\frac{8}{0.8192}\approx9.77\) cm

Answer:

The other acute angle is \(55^{\circ}\), the length of the opposite side is \(5.60\) cm and the length of the hypotenuse is \(9.77\) cm.