Question

given the right triangle shown below with one non - right angle of 39° and an opposite side of length 8: the measure of the other non - right angle is and the lengths of the other sides are: a ≈ c ≈ round your answers to one decimal place. be sure to include the degree symbol with any angle measured in degrees. question help: video submit question

Explanation:

Step1: Find the other non - right angle

In a right triangle, the sum of the interior angles is \(180^{\circ}\), and one angle is \(90^{\circ}\), one angle is \(39^{\circ}\). Let the other non - right angle be \(x\). Then \(x + 90^{\circ}+39^{\circ}=180^{\circ}\), so \(x = 180^{\circ}-90^{\circ}-39^{\circ}=51^{\circ}\).

Step2: Find side \(a\) (adjacent to \(39^{\circ}\))

We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), for \(\theta = 39^{\circ}\), the opposite side is \(8\) and the adjacent side is \(a\). So \(\tan(39^{\circ})=\frac{8}{a}\), then \(a=\frac{8}{\tan(39^{\circ})}\). Since \(\tan(39^{\circ})\approx0.8098\), \(a=\frac{8}{0.8098}\approx9.9\).

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

We know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), for \(\theta = 39^{\circ}\), the opposite side is \(8\) and the hypotenuse is \(c\). So \(\sin(39^{\circ})=\frac{8}{c}\), then \(c = \frac{8}{\sin(39^{\circ})}\). Since \(\sin(39^{\circ})\approx0.6293\), \(c=\frac{8}{0.6293}\approx12.7\).

Answer:

The measure of the other non - right angle is \(51^{\circ}\), \(a\approx9.9\), \(c\approx12.7\)