Find the value of X and Y in the following right triangle. **Title: Solving Right Triangles****Introduction:**In this lesson, we will explore the properties of a right-angled triangle and the methods to solve for unknown sides using trigonometric principles.**Diagram Analysis:**The diagram shows a right-angled triangle with the following properties:- There is a right angle (\(90^\circ\)) at the vertex opposite the hypotenuse.- One of the angles is \(30^\circ\).- The hypotenuse has a length of \(6\sqrt{3}\).- The sides opposite and adjacent to the \(30^\circ\) angle are labeled \(x\) and \(y\), respectively.**Step-by-Step Solution:**1. **Identify Known Values:** - Hypotenuse (\(c\)): \(6\sqrt{3}\) - Angle (\(\theta\)): \(30^\circ\)2. **Use the Sine and Cosine Functions:** The sine function for \(\theta = 30^\circ\): \[ \sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{6\sqrt{3}} \] We know that \(\sin(30^\circ) = \frac{1}{2}\), thus: \[ \frac{1}{2} = \frac{x}{6\sqrt{3}} \] Solving for \(x\): \[ x = \frac{1}{2} \times 6\sqrt{3} = 3\sqrt{3} \] The cosine function for \(\theta = 30^\circ\): \[ \cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{y}{6\sqrt{3}} \] We know that \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), thus: \[ \frac{\sqrt{3}}{2} = \frac{y}{6\sqrt{3}} \] Solving for \(y\): \[ y = \frac{\sqrt{3}}{2} \times 6\sqrt{3} = 3 \times 3 = 9 ### Right Triangle with a 30° AngleThe diagram shown is a right triangle with one of its angles measuring 30 degrees. The triangle includes the following components:1. **Right Angle**: The triangle has one 90° angle, indicated by the small square on the left-hand side of the triangle.2. **30° Angle**: One of the acute angles in the triangle measures 30°, as labeled on the diagram.3. **Legs of the Triangle**: - The leg adjacent to the 30° angle is labeled as **x**. - The leg opposite the 30° angle is labeled as **y**.4. **Hypotenuse**: The hypotenuse, which is the side opposite the right angle and the longest side of the triangle, is labeled as \(5\sqrt{3}\).### Explanation of the DiagramIn trigonometry, a 30-60-90 triangle has a standardized ratio for the lengths of its sides. When the shortest side (adjacent to the 30° angle) is known, the lengths of the other sides can be determined using this ratio: - Hypotenuse: \( 2 \times (\text{shortest side}) \)- The side opposite the 30° angle is half of the hypotenuse.- The side opposite the 60° angle (next to the 30° angle) is \( \sqrt{3} \times (\text{shortest side}) \).Given that the hypotenuse (\(5\sqrt{3}\)) relates to this standard ratio, the triangle's measurements conform to these principles.

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7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT