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**: - **Angle of Elevation**: The angle formed by the line of sight when looking up at an object. - **Angle of Depression**: The angle formed by the line of sight when looking down at an object. ### 2. Solving Bearing Problems - **Bearings**: Bearings are measured in degrees from the north direction (clockwise). - For example, a bearing of \( 30^\circ \) means \( 30^\circ \) clockwise from north. - **Steps to Solve**: 1. Draw a diagram to represent the situation. 2. Mark the bearings and angles clearly. 3. Use trigonometric ratios to find unknown distances or angles. ### 3. Finding Angles of Elevation and Depression - **Steps to Solve**: 1. Identify the height of the object and the distance from the observer. 2. Use the tangent function: - For angle of elevation: \( \tan(\theta) = \frac{\text{height}}{\text{distance}} \) - For angle of depression: The same formula applies, but the height will be negative if you consider the observer's line of sight. ### 4. Simple Three-Dimensional Problems - **Understanding 3D Geometry**: Familiarize yourself with the concepts of 3D shapes, coordinates, and distances. - **Steps to Solve**: 1. Identify the coordinates of the points involved. 2. Use the distance formula in 3D: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] 3. Apply trigonometric ratios to find angles between lines or planes. ### Example Problem **Problem**: A tower is 50 meters tall. From a point on the ground, the angle of elevation to the top of the tower is \( 30^\circ \). Find the distance from the point to the base of the tower. **Solution**: 1. Let \( h = 50 \) meters (height of the tower). 2. Let \( d \) be the distance from the point to the base of the tower. 3. Using the tangent function: \[ \tan(30^\circ) = \frac{h}{d} \] \[ \tan(30^\circ) = \frac{50}{d} \] 4. Rearranging gives: \[ d = \frac{50}{\tan(30^\circ)} \] 5. Calculate \( d \) using the known value of \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \): \[ d = 50 \cdot \sqrt{3} \approx 86.6 \text{ meters} \] ### Conclusion By following these steps and understanding the principles of trigonometry, you can effectively solve problems related to bearings, angles of elevation and depression, and three-dimensional geometry. Practice with various problems to strengthen your skills.