Question

find the value of x and y. 27 30°

Explanation:

Step1: Identify triangle type and trigonometric ratios

This is a right - triangle with one angle \(30^{\circ}\). For a right - triangle, we can use trigonometric ratios (\(\sin\), \(\cos\), \(\tan\)) and the properties of \(30 - 60-90\) triangles. In a \(30 - 60 - 90\) triangle, the side opposite \(30^{\circ}\) is the shortest side, the hypotenuse is twice the side opposite \(30^{\circ}\), and the side opposite \(60^{\circ}\) is \(\sqrt{3}\) times the side opposite \(30^{\circ}\). Also, we can use \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) and \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, the side of length \(27\) is opposite the \(30^{\circ}\) angle? Wait, no. Wait, the right - angle is between the side of length \(27\) and the side \(y\), and the angle of \(30^{\circ}\) is at the end of side \(y\). So the side of length \(27\) is opposite the \(30^{\circ}\) angle? Wait, no. Wait, in a right - triangle, the angle of \(30^{\circ}\): the side opposite \(30^{\circ}\) is the side with length \(27\)? Wait, no, let's re - examine. The right angle is at the vertex where the sides of length \(27\) and \(y\) meet. So the angle of \(30^{\circ}\) is at the vertex of side \(y\) and hypotenuse \(x\). So the side opposite \(30^{\circ}\) is the side with length \(27\), the side adjacent to \(30^{\circ}\) is \(y\), and the hypotenuse is \(x\).

In a \(30 - 60 - 90\) triangle, the hypotenuse \(x = 2\times\) (side opposite \(30^{\circ}\)). Since the side opposite \(30^{\circ}\) is \(27\), then \(x=2\times27 = 54\).

Step2: Find the value of \(y\)

We can use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 27\), \(c=x = 54\), and \(b = y\). So \(y=\sqrt{x^{2}-27^{2}}=\sqrt{54^{2}-27^{2}}=\sqrt{(54 - 27)(54 + 27)}=\sqrt{27\times81}=\sqrt{27}\times\sqrt{81}=9\sqrt{27}=9\times3\sqrt{3}=27\sqrt{3}\). Alternatively, we can use \(\tan(30^{\circ})=\frac{27}{y}\), so \(y=\frac{27}{\tan(30^{\circ})}\). Since \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\), then \(y = 27\sqrt{3}\). Also, using the property of \(30 - 60 - 90\) triangles, the side adjacent to \(30^{\circ}\) (which is \(y\)) is \(\sqrt{3}\) times the side opposite \(30^{\circ}\) (which is \(27\)), so \(y = 27\sqrt{3}\).

Answer:

\(x = 54\), \(y=27\sqrt{3}\)