how to solve questions related to the topic 'applications of trigonometry' \& solving bearing problems, finding the angles of elevation and depression, simple three-dimensional problems.

To solve questions related to the applications of trigonometry, including bearing problems, angles of elevation and depression, and simple three-dimensional problems, follow these steps: ### 1. Understanding the Basics - **Trigonometric Ratios**: Familiarize yourself with sine, cosine, and tangent ratios: - \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \) - \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \) - \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \) - **Angles of Elevation and Depression**: - The angle of elevation is the angle formed by the line of sight when looking up from a horizontal line. - The angle of depression is the angle formed by the line of sight when looking down from a horizontal line. - **Bearings**: Bearings are measured in degrees from the north direction, moving clockwise. For example, a bearing of 30° means 30° east of north. ### 2. Solving Problems Step by Step #### A. Angles of Elevation and Depression 1. **Identify the Right Triangle**: Draw a diagram to visualize the problem. Label the height, distance, and angles. 2. **Use Trigonometric Ratios**: - For angles of elevation, use \( \tan(\theta) = \frac{\text{height}}{\text{distance}} \). - For angles of depression, the same formula applies, but the height is above the horizontal line. 3. **Solve for Unknowns**: Rearrange the formula to find the unknown angle or distance. #### B. Bearing Problems 1. **Draw a Diagram**: Represent the points and angles on a coordinate system. 2. **Use Trigonometry**: Apply the sine and cosine rules if necessary, especially in non-right triangles. 3. **Calculate Distances and Angles**: Use the appropriate trigonometric ratios to find the required distances or angles. #### C. Three-Dimensional Problems 1. **Visualize the Problem**: Draw a 3D diagram if possible, labeling all known dimensions and angles. 2. **Use 3D Trigonometry**: Apply the Pythagorean theorem in three dimensions or trigonometric ratios to find unknown lengths or angles. 3. **Break Down the Problem**: Often, you can break a 3D problem into two or more 2D problems. ### 3. Practice Problems - Solve various problems to strengthen your understanding. Look for problems that require you to find angles of elevation and depression, bearings, and 3D distances. ### Example Problem **Problem**: A tower stands 50 meters tall. From a point 100 meters away from the base of the tower, find the angle of elevation to the top of the tower. **Solution**: 1. **Identify the Triangle**: The height of the tower is the opposite side, and the distance from the tower is the adjacent side. 2. **Use the Tangent Ratio**: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{50}{100} \] 3. **Calculate the Angle**: \[ \theta = \tan^{-1}\left(\frac{50}{100}\right) \] You can use a calculator to find \( \theta \). By following these steps and practicing regularly, you will become proficient in solving problems related to the applications of trigonometry.