📐 Level 1 - Topic 12: Basic Applications of Trigonometry (Heights & Distances) 🚀🌍

1) Trigonometry in Action - Measuring the World Around Us 🌍📏

Welcome to the final topic of Level 1 Trigonometry! In this topic, we'll focus on the practical applications of trigonometry, specifically how it is used to determine heights and distances in various real-world scenarios. You've already started exploring word problems in Topic 10, and now we'll delve deeper into common application types, solidifying your skills in using trigonometry to solve tangible problems.

Why Focus on Heights and Distances?

Core Applications: Finding heights and distances are some of the most fundamental and widely used applications of trigonometry. Historically, trigonometry was developed for these very purposes (e.g., in surveying and navigation).
Intuitive Understanding: Height and distance problems are often easy to visualize and relate to everyday experiences, making them excellent for building intuition about trigonometry.
Foundation for Advanced Applications: Mastering these basic applications is crucial for understanding more complex applications of trigonometry in fields like engineering, physics, and computer graphics.

2) Recap: Problem-Solving Strategy - Your Toolkit 🛠️

Let's quickly revisit the step-by-step strategy we discussed in Topic 10 for solving trigonometry word problems. This strategy is your essential toolkit for tackling height and distance problems:

Understand the Problem: Read carefully, identify knowns and unknowns.
Draw a Diagram: Visualize, sketch, and label a right triangle representing the situation.
Identify Right Triangle & Angle: Locate the right triangle and relevant angle (angle of elevation, depression, etc.).
Choose Trigonometric Ratio: Select sine, cosine, tangent based on known and unknown sides (SOH-CAH-TOA).
Set up Equation: Write the trigonometric equation.
Solve for Unknown: Use algebra and a calculator (or special triangles).
State Answer with Units: Include correct units in your final answer.
Check for Reasonableness: Does your answer make sense in the real-world context?

By consistently following these steps, you can approach even complex word problems systematically and increase your chances of finding the correct solution.

3) Application Type 1: Height Problems - Using Angle of Elevation ⬆️🌲

Many problems involve finding the height of an object (like a tree, building, mountain, flagpole, etc.) using the angle of elevation. We measure the angle from a point on the ground to the top of the object and the distance from that point to the base of the object.

Typical Scenario: You are standing at a certain distance from the base of a tall object. You measure the angle of elevation to its top. You want to find the height of the object.

Example 1: Height of a Building

From a point 80 feet from the base of a building, the angle of elevation to the top of the building is \( 65^\circ \). Find the height of the building.

Solution Steps:

Understand: Distance from base (adjacent) = 80 ft, angle of elevation = \( 65^\circ \), find height (opposite).
Diagram: Right triangle with horizontal base 80 ft, vertical height 'h', angle at base 65°.
Ratio: Tangent (\( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \)).
Equation: \( \tan(65^\circ) = \frac{h}{80} \).
Solve: \( h = 80 \times \tan(65^\circ) \approx 80 \times 2.1445 \approx 171.56 \) feet.
Answer: The building is approximately 171.56 feet tall.

4) Application Type 2: Distance Problems - Using Angle of Depression ⬇️🚢

Another common type is finding horizontal distances using the angle of depression. Often, you are at a height (e.g., on a cliff, building, airplane) and looking down at an object at a lower level.

Typical Scenario: You are at a known height above the ground/sea level. You measure the angle of depression to an object below. You want to find the horizontal distance to the object.

Example 2: Distance to a Boat

From the top of a lighthouse 150 feet above sea level, the angle of depression to a boat is \( 30^\circ \). How far is the boat from the base of the lighthouse? (Assume the base is at sea level).

Solution Steps:

Understand: Lighthouse height (opposite) = 150 ft, angle of depression = \( 30^\circ \), find horizontal distance to boat (adjacent).
Diagram: Right triangle with vertical height 150 ft, horizontal distance 'd', angle of elevation from boat to top is 30°.
Ratio: Tangent (\( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \)).
Equation: \( \tan(30^\circ) = \frac{150}{d} \).
Solve: \( d = \frac{150}{\tan(30^\circ)} = \frac{150}{(1/\sqrt{3})} = 150\sqrt{3} \approx 259.81 \) feet.
Answer: The boat is approximately 259.81 feet from the base of the lighthouse.

5) Application Type 3: Problems with Ladders and Inclined Planes 🪜

Problems involving ladders leaning against walls or objects on inclined planes often use trigonometry to find heights, distances, or angles. These scenarios naturally form right triangles.

Typical Scenarios:

Ladder leaning against a wall: Ladder is hypotenuse, wall and ground are legs.
Ramp or inclined plane: Ramp/plane is hypotenuse, vertical rise and horizontal run are legs.

Example 3: Ladder Height

A 20-foot ladder leans against a building, making an angle of \( 70^\circ \) with the ground. How high up the wall does the ladder reach?

Solution Steps:

Understand: Ladder length (hypotenuse) = 20 ft, angle with ground = \( 70^\circ \), find wall height (opposite).
Diagram: Right triangle, hypotenuse 20 ft (ladder), vertical height 'h', angle at ground 70°.
Ratio: Sine (\( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)).
Equation: \( \sin(70^\circ) = \frac{h}{20} \).
Solve: \( h = 20 \times \sin(70^\circ) \approx 20 \times 0.9397 \approx 18.79 \) feet.
Answer: The ladder reaches approximately 18.79 feet up the wall.

6) Application Type 4: Navigation and Distance Between Points 🧭

Trigonometry is fundamental in navigation. While more advanced navigation uses spherical trigonometry, basic trigonometry helps solve problems involving directions (North, South, East, West) and distances traveled in different directions. We can find straight-line distances and resultant directions.

Typical Scenarios:

Traveling in cardinal directions (East then North, etc.): Use Pythagorean theorem and tangent to find distance and direction.
Traveling on bearings (simplified bearings from North or East/West): Break down into North/South and East/West components using sine and cosine.

Example 4: Distance and Direction from Starting Point

A person walks 3 km due East, then 4 km due South. How far and in what direction is the person from their starting point?

Solution Steps:

Understand: Walks East 3km, then South 4km, find straight-line distance and direction from start.
Diagram: Right triangle, East leg 3km, South leg 4km, hypotenuse 'd', angle \( \theta \) East of South.
Distance: Pythagorean Theorem: \( d^2 = 3^2 + 4^2 = 9 + 16 = 25 \), \( d = \sqrt{25} = 5 \) km.
Direction: Tangent: \( \tan(\theta) = \frac{\text{Opposite (East)}}{\text{Adjacent (South)}} = \frac{3}{4} \), \( \theta = \arctan\left(\frac{3}{4}\right) \approx 36.87^\circ \). Direction is approximately \( 36.87^\circ \) East of South (or S \( 36.87^\circ \) E).
Answer: The person is 5 km from the start, in a direction about \( 36.87^\circ \) East of South.

7) Practice Questions 🎯

7.1 Fundamental – Basic Height and Distance Applications

1. A tree casts a shadow of 20 meters when the angle of elevation of the sun is \( 40^\circ \). Find the height of the tree.

2. A 30-foot ladder is leaned against a wall so that it reaches 24 feet up the wall. What angle does the ladder make with the ground?

3. From the top of a cliff 120 meters high, the angle of depression to a swimmer is \( 20^\circ \). How far is the swimmer from the base of the cliff?

4. A kite is flying at an angle of elevation of \( 55^\circ \). If 80 meters of string are out, how high is the kite above the ground?

5. A person walks 6 km due North and then 2.5 km due East. How far is the person from the starting point?

6. A ramp needs to have an angle of elevation of \( 10^\circ \). If the horizontal distance covered by the ramp is 5 meters, what is the vertical height (rise) of the ramp?

7. From a point on the ground 50 feet from a building, the angle of elevation to the top of a flagpole on top of the building is \( 68^\circ \), and the angle of elevation to the top of the building is \( 60^\circ \). Find the height of the flagpole.

8. An airplane takes off at an angle of \( 8^\circ \) and travels at a speed of 250 km/h. What is its altitude after 15 minutes of flight?

9. A surveyor sights a point directly across a river. She then walks 100 meters along the riverbank and finds that the angle to the same point is now \( 35^\circ \) (angle of sightline relative to the riverbank). How wide is the river?

10. True or False: In solving height and distance problems, drawing a diagram is an optional step and not essential. Explain.

7.2 Challenging – Applied & Multi-Concept Problems 💪🚀

1. Two buildings are 100 meters apart. From the top of the shorter building, the angle of elevation to the top of the taller building is \( 25^\circ \), and the angle of depression to the base of the taller building is \( 40^\circ \). Find the height of the taller building.

2. A hiker starts at point A and walks 4 km in the direction N \( 40^\circ \) E to point B. Then, from B, the hiker walks 6 km due East to point C. Find the straight-line distance from A to C and the bearing of C from A.

3. A lighthouse is situated on a cliff 200 feet above sea level. From a boat, the angle of elevation to the top of the lighthouse is \( 28^\circ \), and the angle of elevation to the top of the cliff is \( 20^\circ \). Find the height of the lighthouse.

4. A security camera is mounted on a wall 9 feet above the ground. It needs to cover a horizontal distance of 15 feet out from the base of the wall. What is the minimum angle of depression needed for the camera to cover this area?

5. (Conceptual) Design a practical scenario in your daily life where you could use trigonometry to indirectly measure a height or distance that would be difficult to measure directly with a ruler or measuring tape. Describe the scenario, the measurements you would take, and how you would use trigonometry to find the unknown height or distance.

8) Summary - Level 1 Trigonometry Applications 🎉

Heights and Distances: Core applications of trigonometry in real-world measurement.
Problem-Solving Strategy: Essential steps: Understand, Diagram, Ratio, Equation, Solve, Answer, Check.
Angle of Elevation: Upward angle to object above horizontal - used for height problems.
Angle of Depression: Downward angle to object below horizontal - used for distance problems.
Ladders, Ramps, Inclines: Common scenarios forming right triangles for height/distance calculations.
Basic Navigation: Directions and distances, using trigonometry in simplified navigation contexts.
Practice Makes Perfect: Solving various types of height and distance problems is key to mastering these applications.

Congratulations on completing Level 1 Trigonometry - Basic Foundations! You've come a long way, from understanding angles and trigonometric ratios to applying trigonometry to solve real-world problems involving heights and distances. You've built a solid foundation in trigonometry. Keep practicing and exploring, and you'll find trigonometry to be a powerful tool in mathematics and beyond! 🚀🌍🌟 Ready to move on to Level 2 and explore more advanced trigonometry?