Apply the Laws of Sines and Cosines

Lesson 1 of 2: Surveying and Navigation

In this lesson:

• Use Law of Sines and Cosines to solve real-world triangle problems
• Apply the 5-step framework to surveying and navigation scenarios
• Verify that answers are reasonable in context

Learning Objectives

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

1. Apply the Law of Sines to find unknown measurements in triangles
2. Apply the Law of Cosines to find unknown measurements in triangles
3. Solve real-world problems involving surveying and navigation
4. Verify solutions and assess the reasonableness of answers in context

From Proof to Practice

You know the Laws of Sines and Cosines hold for any triangle. Now the question is: how do you apply them when the triangle is hidden inside a real problem?

• A canyon you can't cross
• A ship navigating by compass
• A landmark too far to measure directly

Every one of these is a triangle problem in disguise.

The 5-Step Framework

Use this process for every applied triangle problem — surveying, navigation, forces, or otherwise.

Surveying: Measuring the Unmeasurable

Surveyors measure distances and boundaries that can't be measured directly — across rivers, canyons, or to landmarks that can't be reached.

The key technique: triangulation

• Measure a baseline — a distance you can measure
• Measure angles from each end of the baseline to the target
• Form a triangle and solve it

Surveying Setup

A baseline connects points A and C. Angles at A and C are measured to target B.

Step 1 Is Non-Negotiable: Draw the Diagram

Before writing a single equation:

• Sketch the situation — even rough is fine
• Label every known quantity (sides and angles)
• Mark what you're solving for with a "?"

Why it matters: The diagram shows you the triangle structure. Without it, you can't identify the case, and you can't set up the correct law.

Most applied-problem errors start with skipping the diagram.

Worked Example: The Canyon

Given: Baseline AC = 50 m, angle at A = 70°, angle at C = 60°.
Find: The distance AB across the canyon.

Step 1: Draw the diagram (done — see Slide 6)
Step 2: Find the missing angle

Step 3: Identify the case — two angles and the included side → ASA
Step 4: Which law? → Law of Sines

Worked Example: The Canyon (Solution)

Law of Sines:

The canyon is approximately 56.6 meters wide.

Quick Check

A triangle has two known angles (55° and 75°) and the side between them is 80 m.

Which case is this? Which law do you use?

Think before advancing...

Always Label Units — Always Verify

Units rule:

• Label every quantity: 56.6 m, not just 56.6
• Convert to consistent units before calculating
• Never mix meters and kilometers in the same calculation

Reasonableness check (Step 5 of the framework):

• Is the magnitude in the right range for this scenario?
• Does it satisfy triangle inequality?
• Do angles sum to 180°?

Worked Example: River Crossing

A surveyor stands at A on one bank. She measures angle to a tree B on the far bank: 65°. She walks 100 m along the shore to C and measures angle to B: 80°.

Find: Distance AB (the river width).

Setup:

• Triangle ABC: AC = 100 m, angle A = 65°, angle C = 80°
• Find angle B:
• Case: ASA → Law of Sines

Worked Example: River Crossing (Solution)

The river is approximately 172 meters wide.

Reasonableness check: is 172 m a reasonable river width? ✓

Your Turn: Triangulate a Landmark

A hiker at point A measures the angle to a distant peak B as 40°.
She walks 200 m to point C along the baseline.
From C, she measures the angle to B as 65°.

1. Draw the triangle — label all known values
2. Identify the case
3. Set up (but don't solve) the Law of Sines equation for AB

Try it, then advance for the setup

Is This Answer Reasonable?

Using the Law of Sines for the landmark problem gives:

Is 187.7 meters reasonable?

• Angle B = 180° − 40° − 65° = 75° ✓ (angles sum to 180°)
• AB < baseline × 2 for this geometry ✓
• A landmark 188 m away — visible and measurable with instruments ✓

Surveying Practice

A surveyor needs to find the distance from point A to a tower B across a marsh. She measures the baseline AC = 80 m. From A, the angle to B is 55°. From C, the angle to B is 68°.

Find: Distance AB.

Show your full work: draw, identify case, apply law, verify.

From Land to Sea: Navigation

Surveyors work on land. Navigators work at sea and in the air — and they face the same triangle math.

New concept: bearings

• A bearing is an angle measured clockwise from north
• 000° = North · 090° = East · 180° = South · 270° = West
• 060° = 60° clockwise from north (northeast)

Bearings replace compass directions in navigation problems.

Reading a Compass: Bearings

Bearings: always clockwise from north, always three digits (000°–360°).

Converting Bearings to Triangle Angles

The bearing is not the triangle's interior angle.

How to find the angle between two paths:

If Ship A travels at bearing and Ship B at bearing :

If the result > 180°, subtract from 360°.

Example: bearings 030° and 120° → angle = 120° − 030° = 90°

Worked Example: Two Ships

Ship A leaves port at bearing 030° and travels 40 km.
Ship B leaves port at bearing 120° and travels 50 km.

Find: How far apart are the ships?

Setup:

• Angle between paths: 120° − 030° = 90°
• Triangle: two sides 40 km and 50 km, included angle 90°
• Case: SAS with a right angle → Pythagorean Theorem

Worked Example: Two Ships (Solution)

Since the included angle is 90°:

If the angle weren't 90°, we'd use the Law of Cosines:

Quick Check

Two ships leave a port. Ship A travels at bearing 045°. Ship B travels at bearing 150°.

What is the interior angle between their paths?

Think before advancing...

Worked Example: Non-Right Navigation

Two ships leave port. Ship A travels 60 km at bearing 040°. Ship B travels 75 km at bearing 160°.

Angle between paths: 160° − 040° = 120°

This is SAS with a non-right included angle → Law of Cosines:

Your Turn: Navigation Diagram

A plane leaves an airport at bearing 050° and flies 120 km. It then changes course to bearing 170° and flies 90 km.

1. Draw the flight path on a compass diagram
2. Find the angle between the two legs
3. Identify the case and which law to use

Don't solve yet — just set up

Quick Check

In the navigation problem above, the angle between the two flight legs is 120° and the two sides are 120 km and 90 km.

Which law applies, and why?

Think before advancing...

Navigation Practice

Two planes leave an airport simultaneously. Plane A flies at 25 km/h on bearing 030° for 2 hours. Plane B flies at 30 km/h on bearing 120° for 2 hours.

Find: How far apart are the planes after 2 hours?

Show full work: draw, label, identify case, apply law, verify

Key Takeaways

✓ The 5-step framework works for every applied triangle problem
✓ Law of Sines: two angles + one side (AAS or ASA)
✓ Law of Cosines: two sides + included angle (SAS) or three sides (SSS)
✓ Surveying: baseline + two angles → ASA → Law of Sines
✓ Navigation: convert bearings to interior angles, then identify the case

Draw the diagram first — no diagram, no reliable equation
Bearings ≠ triangle angles — subtract bearings to get the interior angle
Wrong law — always identify AAS/ASA/SAS/SSS before choosing
Missing units — label every measurement; answers without units are incomplete
Skip verification — always ask: is this answer reasonable in context?

What's Next: Lesson 2

Coming up in Lesson 2:

• Resultant forces — vectors added tip-to-tail, triangle math
• Ambiguous case (SSA) — when two solutions, one solution, or no solution exist
• Multi-step problems — breaking complex scenarios into solvable triangles

The same framework, applied to more complex situations.