Apply Laws of Sines and Cosines

Lesson 2 of 2: Forces and Advanced Problems

In this lesson:

• Find resultant forces using triangle methods
• Handle the ambiguous case (SSA)
• Break multi-step problems into solvable triangles

Learning Objectives

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

4. Calculate resultant forces and resolve vectors using triangle methods
5. Handle the ambiguous case (SSA) in applied contexts
6. Choose appropriate strategies for complex multi-step problems
7. Verify solutions and assess the reasonableness of answers in context

Same Laws, New Context

In Lesson 1: surveyors measured distances and navigators plotted courses.

Today: engineers calculate forces — the push-and-pull combinations that move objects.

The triangle is the same computational tool. The numbers now carry units of Newtons.

Forces as Vectors

A force has two essential properties:

• Magnitude — how strong (measured in Newtons, N)
• Direction — which way it acts (measured as an angle)

When two forces act on one object, their combined effect is the resultant — itself a vector with a magnitude and direction.

Tip-to-Tail Vector Addition

F1 and F2 placed tip-to-tail form a triangle — the resultant R closes it.

The Interior Angle Rule

where is the tail-to-tail angle between the two forces.

• → interior angle = 120°
• → interior angle = 90° (right triangle)
• → interior angle = 180° (collinear — no triangle)

Worked: Right-Angle Forces (Setup)

Given: F1 = 40 N east, F2 = 60 N north. Angle between forces: 90°.

Interior angle = 180° − 90° = 90° → the force triangle has a right angle.

When the interior angle is 90°, the Law of Cosines reduces to the Pythagorean Theorem.

Right-Angle Forces: Solution

• Direction: north of east
• R is larger than either force alone — and smaller than 40 + 60 = 100 N ✓

Quick Check

Two forces are 60° apart (measured tail-to-tail).

What is the interior angle of the force triangle, and why is it not 60°?

Think before advancing...

Non-Right Forces: Law of Cosines

When forces are not 90° apart, the triangle is SAS:

• Two sides = force magnitudes (, )
• Included angle = interior angle =

Use the Law of Cosines for the resultant magnitude:

Worked: F1 = 50 N, F2 = 80 N at 60° (Setup)

Given: F1 = 50 N, F2 = 80 N, angle between them = 60°.

Interior angle = 180° − 60° = 120°

Law of Cosines (SAS — two magnitudes + included interior angle):

Forces at 60°: Solution

Direction (Law of Sines): from F1.

Resultant: 113.6 N at 37.6° from F1 toward F2 — always label the units.

Your Turn: Force Problem

Given: F1 = 30 N, F2 = 45 N, angle between them = 80°.

1. Compute the interior angle of the force triangle
2. Identify the case (SAS) and write the Law of Cosines for
3. Write the full equation — do not solve yet

Try before advancing...

Quick Check

In the force problem above (F1 = 30 N, F2 = 45 N, angle 80°):

The interior angle is 100°. Is positive or negative — and what does that mean for ?

Think before advancing...

Stepping Up the Challenge

Force problems involve one clear triangle with a known case.

Two harder situations require extra care:

• SSA (ambiguous case): may have 0, 1, or 2 valid triangles
• Multi-step problems: results from one triangle feed the next

Both need strategy before calculation.

The Ambiguous Case: SSA

SSA — two sides and a non-included angle — may produce:

• 0 solutions — side is too short to reach the base line
• 1 solution — exactly one valid triangle (or a right triangle)
• 2 solutions — two distinct triangles both satisfy the given data

Always check before solving — the next slide shows the decision method.

SSA Decision Flowchart

Compute ; compare , , in order to find the solution count.

Worked: SSA Surveyor Example

Given: m, m, angle

Check: m

Since (106.1 < 120 < 150): two solutions exist

Quick Check

An SSA problem: , , angle .

How many solutions exist? Show the comparison.

Compute , then place in the correct zone.

Multi-Step Problems: The Framework

Complex problems contain multiple triangles. Solve in order:

1. Draw — one complete diagram before any equations
2. Label — mark all knowns and unknowns
3. Plan — identify which triangle you can solve first
4. Solve — use each result in the next triangle
5. Verify — does the answer make physical sense?

Worked: Two-Observer Balloon (Setup)

Setup: A and B are 100 m apart on level ground. The balloon is directly above a point between them.

• From A: angle of elevation = 60°
• From B: angle of elevation = 45°

Let = horizontal distance from A to the point below the balloon; then = distance from B.

Balloon: Two Equations, Two Unknowns

From A (angle of elevation 60°):

From B (angle of elevation 45°):

Balloon: Solving for Height

Substitute into :

Your Turn: Similar Setup

From point A, the angle of elevation to a rooftop is 10°. From point B, 50 m closer, the angle is 15°. The building is 25 m tall.

Task: Draw the diagram. Identify the two right triangles. Label all unknowns.

Do not calculate yet — just set up the structure.

Quick Check

If = horizontal distance from B to the building base, which equation is correct?

• A:
• B:

Think before advancing...

Practice: Ranger Stations

Two stations, A and B, are 20 km apart. A fire is spotted:

• From A at bearing 070°
• From B at bearing 340°

Find the distance from each station to the fire.

Hint: Convert bearings to interior angles of triangle ABF, then apply Law of Sines (AAS).

Key Takeaways

✓ Force resultant: interior angle = 180° − α → Law of Cosines (SAS)
✓ SSA: compute ; compare , , to count solutions
✓ Multi-step: draw → plan → solve → verify in order
✓ Check physical reasonableness — 63 m balloon? Plausible.

SAS → Law of Cosines, not Law of Sines
SSA → run the -check first — never assume one solution

You Can Solve Any Triangle Problem

Apply to any context — land, sea, forces, engineering.