Applications

Apply trigonometry to bearing and distance problems. Math 20-1 Alberta.

Lesson 4.7 of Trigonometry in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Choosing the Right Law. Review the decision rule for when to apply the Cosine Law versus the Sine Law, and learn a five-step strategy for turning any word problem into a solvable triangle.
• Distance and Surveying Problems. Apply the Sine Law to two surveying problems: find the distance to a remote point given two angles and a baseline, and find a triangle's perimeter when only a relationship between sides is known.
• Real-World Geometry Problems. Use the Cosine Law and mixed oblique-plus-right-triangle techniques to find the distance a racquetball travels and the height of a building using two angles of elevation.

Inside the lesson — a free preview

Quick Recap — Which Law?

When to Use Each Law. Before applying any formula, identify which information you already have about the triangle.

Cosine Law — use when you have:

Two sides and the angle between them (SAS), or

All three sides (SSS).

Sine Law — use when you have:

A matching pair (an angle and its opposite side) plus one more piece of information, or

Two angles and any side (the third angle follows from the angle sum).

In Short. Use the Cosine Law in the SSS or SAS cases. Use the Sine Law in all other cases.

Strategy for Word Problems

Five-Step Problem-Solving Strategy. Step 1 — Draw a diagram.

Label all known sides and angles clearly. A good diagram prevents most errors.

Step 2 — Identify the triangle.

Name the three vertices. Write down every side length and angle you know.

Step 3 — Decide which law applies.

Check whether you have SAS, SSS (Cosine Law), or a matching pair plus one more piece (Sine Law).

Step 4 — Set up and solve.

Write the chosen formula, substitute the known values, and solve algebraically.

Step 5 — Check your answer.

Confirm the result is reasonable in context. A building height should not be millions of metres; a distance between two points should not be negative.

Worked example: Find the Distance to a Store

Problem — Two Cabins and a Store. Two cabins, A and B, are 1.8 km apart. The angle at Cabin A is 61° and the angle at Cabin B is 88°. There is a store at point C. How far is the store from Cabin A, to the nearest tenth?

Solution — Example 1. Step 1: Identify what we have.

We know two angles (A = 61°, B = 88°) and one side (c = 1.8 km, the side opposite C). We want side b (from A to C, opposite B). Two angles and a side — use the Sine Law.

Step 2: Find the missing angle C.

A + B + C = 180°

61° + 88° + C = 180°

C = 180° - 61° - 88° = 31°

Step 3: Set up the Sine Law.

Use the matching pair (C, c) — both are known:

(b)/( B) = (c)/( C)

(b)/( 88°) = (1.8)/( 31°)

Step 4: Solve for b.

b = (1.8 · 88°)/( 31°)

Step 5: Compute (degree mode).

88° ≈ 0.9994, 31° ≈ 0.5150

b ≈ (1.8 × 0.9994)/(0.5150) ≈ (1.7989)/(0.5150) ≈ 3.49

Answer: The store is approximately 3.5 km from Cabin A.

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