Solving Right Triangles

Part 2 of 2: Applications

In this lesson:

• Solve problems with angles of elevation and depression
• Apply right triangles to navigation and bearings
• Use the "draw first, solve second" approach

Learning Objectives

By the end of this lesson, you should be able to:

1. Solve real-world problems using angles of elevation and depression
2. Apply right triangle solving to navigation and bearings
3. Draw accurate diagrams before solving

How Tall Is That Building?

You can measure it - without climbing.

Angle of Elevation

Measured upward from the horizontal at eye level

Angle of Depression

Measured downward from the horizontal at eye level

Key Relationship

Elevation from bottom = Depression from top (alternate interior angles)

Check-In

Where is the horizontal reference - on the ground or at eye level?

Think about why this matters for drawing your diagram...

Example 1: Building Height

Given: 50 m from building, 60° elevation

Example 1: Solution

Find: height of building

Example 2: Distance to Boat

Given: 100 m cliff, 25° depression to boat

Example 2: Solution

Find: horizontal distance to boat

Check-In

In Example 2, where exactly is the right angle in your diagram?

Locate it before moving on...

Example 3: Line-of-Sight Distance

Given: airplane at 5000 ft, 40° elevation

Example 3: Solution

Find: line-of-sight distance

Why sine, not tangent? Because we're connecting opposite to hypotenuse.

Key Tips

When solving elevation and depression problems:

1. Draw first, solve second - always
2. Horizontal reference is at eye level, not the ground
3. Label O, A, H relative to the angle you're using
4. Check: Does your answer make sense? (e.g., building height ≈ reasonable)

Transition

Same tools, new context - from looking up and down to navigating across

Bearing Notation

N30°E = "start at north, rotate 30° toward east"

Compass Practice

Practice reading: N30°E, S45°W, N60°W, bearing 090°

Check-In

Draw N40°E on a compass. Which direction do you start from?

Visualize it before moving on...

Example 1: Hiker Distance and Bearing

Given: 4 km east, then 3 km north

Example 1: Solution

Find: distance and bearing

Distance: km

Bearing angle: → N53.1°E

Example 2: Component Breakdown

Given: plane flies N30°E for 200 km

Example 2: Solution

Find: north and east components

North: km

East: km

Check-In

For bearing N30°E - which component uses cosine: north or east?

Think about opposite, adjacent, and hypotenuse...

Example 3: Multi-Leg Journey

Given: 10 km N, 15 km E, 5 km S

Example 3: Solution

Find: displacement distance and bearing

Net: N = 10 − 5 = 5 km, E = 15 km

Distance: km

Bearing: → N71.6°E

Summary

"Draw first, solve second" - right triangles are everywhere

✓ Elevation and depression problems: horizontal reference at eye level
✓ Navigation problems: break into N/S and E/W components
✓ Same tools: trig ratios, inverse trig, Pythagorean Theorem

You can now solve right triangles in the real world.