Question

given the right triangle shown below with one non - right angle of $59^{circ}$ and an opposite side of length 14: the measure of the other non - right angle is \boxed{} and the lengths of the other sides are: $a\approx\boxed{}$ $c\approx\boxed{}$ round your answers to one decimal place. be sure to include the degree symbol with any angle measured in degrees. question help: \boxed{video}

Explanation:

Step1: Find the other non - right angle

In a right triangle, the sum of the non - right angles is \(90^{\circ}\). Let the other non - right angle be \(x\). We know one non - right angle is \(59^{\circ}\), so \(x + 59^{\circ}=90^{\circ}\). Solving for \(x\), we get \(x = 90^{\circ}- 59^{\circ}=31^{\circ}\).

Step2: Find the length of side \(a\) (adjacent to \(59^{\circ}\))

We know the opposite side to \(59^{\circ}\) is \(14\) (the vertical side). We can use the tangent function: \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). For \(\theta = 59^{\circ}\), \(\tan(59^{\circ})=\frac{14}{a}\), so \(a=\frac{14}{\tan(59^{\circ})}\). Since \(\tan(59^{\circ})\approx1.6643\), then \(a=\frac{14}{1.6643}\approx8.4\).

Step3: Find the length of side \(c\) (hypotenuse)

We can use the sine function: \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\theta = 59^{\circ}\), \(\sin(59^{\circ})=\frac{14}{c}\), so \(c = \frac{14}{\sin(59^{\circ})}\). Since \(\sin(59^{\circ})\approx0.8572\), then \(c=\frac{14}{0.8572}\approx16.3\).

Answer:

The measure of the other non - right angle is \(31^{\circ}\), \(a\approx8.4\), \(c\approx16.3\)