Question

given the right triangle shown below with one non - right angle of 30° and an adjacent side of length 14: the measure of the other non - right angle is
and the lengths of the other sides are:
b ≈
c ≈
round your answers to one decimal place.
be sure to include the degree symbol in your answer for any angle measured in degrees.
question help:
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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 \(30^{\circ}\). Let the other non - right angle be \(x\). Then \(x + 90^{\circ}+30^{\circ}=180^{\circ}\), so \(x = 180^{\circ}-90^{\circ}-30^{\circ}=60^{\circ}\).

Step2: Identify the sides

We know that in a right triangle, the side adjacent to the \(30^{\circ}\) angle is \(a = 14\) (the vertical leg), the side opposite to the \(30^{\circ}\) angle is \(b\), and the hypotenuse is \(c\).

Using the tangent function for the \(30^{\circ}\) angle: \(\tan(30^{\circ})=\frac{b}{a}\), since \(a = 14\) and \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\approx0.577\), then \(b=a\times\tan(30^{\circ})=14\times\frac{1}{\sqrt{3}}\approx14\times0.577\approx8.1\) (rounded to one decimal place).

Using the cosine function for the \(30^{\circ}\) angle: \(\cos(30^{\circ})=\frac{a}{c}\), \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\approx0.866\), so \(c=\frac{a}{\cos(30^{\circ})}=\frac{14}{\frac{\sqrt{3}}{2}}=\frac{28}{\sqrt{3}}\approx\frac{28}{1.732}\approx16.2\) (rounded to one decimal place).

Answer:

The measure of the other non - right angle is \(60^{\circ}\), \(b\approx8.1\), \(c\approx16.2\)