Question

find the length of side ( x ) to the nearest tenth.

answer attempt 2 out of 4
( x = ) submit answer

Explanation:

Step1: Identify the trigonometric ratio

We have a right - triangle with one angle of \(30^{\circ}\), the adjacent side to the \(30^{\circ}\) angle is \(x\) and the opposite side is \(4\)? Wait, no. Wait, the angle of \(30^{\circ}\), the side adjacent to \(30^{\circ}\) is \(x\)? Wait, no. Let's re - examine the triangle. The right - angle, one angle is \(30^{\circ}\), one angle is \(60^{\circ}\). The side with length \(4\) is adjacent to the \(30^{\circ}\) angle, and \(x\) is opposite to the \(30^{\circ}\) angle? Wait, no. Wait, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For the \(30^{\circ}\) angle, \(\tan(30^{\circ})=\frac{x}{4}\)? Wait, no. Wait, if the angle is \(30^{\circ}\), the side opposite to \(30^{\circ}\) is \(x\) and the side adjacent is \(4\)? Wait, no, \(\tan(30^{\circ})=\frac{\text{opposite}}{\text{adjacent}}\), so \(\tan(30^{\circ})=\frac{x}{4}\), then \(x = 4\times\tan(30^{\circ})\). But \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\approx0.577\), so \(x = 4\times0.577\approx2.3\)? Wait, no, maybe we can use \(\tan(60^{\circ})\). Wait, the angle of \(60^{\circ}\), the opposite side is \(4\) and the adjacent side is \(x\). So \(\tan(60^{\circ})=\frac{4}{x}\), since \(\tan(60^{\circ})=\sqrt{3}\approx1.732\). Then \(x=\frac{4}{\tan(60^{\circ})}=\frac{4}{\sqrt{3}}\approx\frac{4}{1.732}\approx2.3\). Alternatively, we can use the 30 - 60 - 90 triangle ratios. In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest side, opposite \(60^{\circ}\) is \(\sqrt{3}\) times the shortest side, and the hypotenuse is twice the shortest side. Wait, if the side adjacent to \(30^{\circ}\) (the longer leg) is \(x\), and the side opposite to \(30^{\circ}\) (the shorter leg) is \(y\), then \(x = y\sqrt{3}\) and hypotenuse \(h = 2y\). Wait, in our triangle, the side with length \(4\) is the longer leg (adjacent to \(30^{\circ}\)), so \(x\) is the shorter leg (opposite to \(30^{\circ}\)). So if the longer leg (adjacent to \(30^{\circ}\)) is \(x\sqrt{3}\), and here the longer leg is \(4\), so \(x\sqrt{3}=4\), then \(x=\frac{4}{\sqrt{3}}\approx2.3\). Wait, let's calculate it properly. \(\frac{4}{\sqrt{3}}=\frac{4\sqrt{3}}{3}\approx\frac{4\times1.732}{3}=\frac{6.928}{3}\approx2.3\).

Step2: Calculate the value

We know that \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\approx0.577\), and if we consider the angle of \(30^{\circ}\), and use the formula \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), where \(\theta = 30^{\circ}\), opposite side is \(x\), adjacent side is \(4\). So \(x = 4\times\tan(30^{\circ})\). \(\tan(30^{\circ})\approx0.577\), so \(x=4\times0.577 = 2.308\approx2.3\) (to the nearest tenth). Or using \(\tan(60^{\circ})=\sqrt{3}\approx1.732\), and for angle \(60^{\circ}\), \(\tan(60^{\circ})=\frac{4}{x}\), so \(x = \frac{4}{\tan(60^{\circ})}=\frac{4}{1.732}\approx2.3\).

Answer:

\(2.3\)