Exercise 8

9.1.8 Exercise 8

Description

Quick Overview

This section presents several exercises involving the calculation of heights and distances using trigonometric principles.

Standard

The exercises in this section explore real-world applications of trigonometry, specifically involving angles of elevation and depression. They require students to calculate heights of objects like poles, towers, and trees using given angles and distances.

Detailed

Exercise 9.1 Detailed Summary

This section focuses on practical applications of trigonometry, particularly through exercises that require calculations involving angles of elevation and depression. Each problem presents a scenario where students must determine unknown heights or distances using given lengths and angles. Important concepts include:

  1. Trigonometric Ratios: Understanding how to apply sine, cosine, and tangent ratios to solve for missing values in right-angled triangles.
  2. Real-World Applications: Exercises such as determining the height of poles, towers, trees, and even the distances traveled by objects in different scenarios (like balloons and kites).
  3. Visual Representation: Each problem is designed to be visualized with diagrams, reinforcing spatial understanding and helping students approach geometry interactively.

Through these exercises, learners gain insights into how trigonometry applies to everyday situations and the process of modeling real-world problems mathematically.

Key Concepts

  • Trigonometric Ratios: Used to find missing side lengths of triangles based on angles.

  • Angle of Elevation: Important in determining height when looking up.

  • Angle of Depression: Vital for calculating height when looking down.

Memory Aids

🎵 Rhymes Time

  • To find a pole's height, so neat, use sine for the rise, and make it complete.

📖 Fascinating Stories

  • Imagine a tree bending with the wind, its top touches the ground and we find its height with a trusty sine.

🧠 Other Memory Gems

  • SOH-CAH-TOA helps remember sin, cos, and tan: Sine is Opposite over Hypotenuse, Cosine is Adjacent, and Tangent is Opposite over Adjacent.

🎯 Super Acronyms

HAT means Height of the pole by Angle and Tangent for calculations.

Examples

  • A circus artist climbs a rope at a 30° angle, allowing us to find the height of the pole using trigonometric functions.

  • A broken tree creates an angle with the ground; using distance to the tree base helps find its height.

Glossary of Terms

  • Term: Trigonometric Ratios

    Definition:

    Ratios derived from the angles of a right-angled triangle, specifically sine, cosine, and tangent.

  • Term: Angle of Elevation

    Definition:

    The angle formed by the line of sight and the horizontal when looking up at an object.

  • Term: Angle of Depression

    Definition:

    The angle formed by the line of sight and the horizontal when looking down at an object.