Mathematics · Grade 11 · Trigonometric Ratios and Functions · Term 3

The Cosine Law

Applying the Cosine Law to solve for unknown sides and angles in non-right triangles.

• Ontario Curriculum Expectations
• HSG.SRT.D.11

About This Topic

The Cosine Law lets students solve for unknown sides and angles in any triangle, extending beyond right triangles. The formula c² = a² + b² - 2ab cos C applies when two sides and the included angle or all three sides are known. Students explain its use compared to the Sine Law, which requires a pair of sides and a non-included angle or angle-side-angle. This addresses key curriculum questions on tool selection and problem design.

Within Ontario's Grade 11 Trigonometric Ratios and Functions unit, the topic strengthens algebraic skills with trigonometry and geometric visualization. Students manipulate equations to isolate variables, compare law requirements, and create real-world scenarios like surveying land or analyzing forces in physics. These connections build problem-solving flexibility for advanced math.

Active learning supports this topic through hands-on construction and collaboration. When students build triangles using rulers, protractors, and string, then verify measurements with the formula, they see the law in action. Group tasks designing application problems encourage discussion of givens and strategies, turning formulas into practical tools students own.

Key Questions

1. Explain when the Cosine Law is the appropriate tool for solving a triangle.
2. Compare the information required to use the Sine Law versus the Cosine Law.
3. Design a real-world problem that requires the application of the Cosine Law.

Learning Objectives

• Calculate the length of an unknown side of a triangle given two sides and the included angle using the Cosine Law.
• Determine the measure of an unknown angle in a triangle given all three side lengths using the Cosine Law.
• Compare the conditions under which the Sine Law and Cosine Law are the most efficient tools for solving triangles.
• Design a word problem that requires the application of the Cosine Law to find a missing side or angle.

Before You Start

Pythagorean Theorem

Why: Students need a foundational understanding of how side lengths relate in right triangles before extending to non-right triangles.

Basic Trigonometric Ratios (SOH CAH TOA)

Why: Familiarity with sine, cosine, and tangent in right triangles is essential for understanding their application in the Cosine Law.

Sine Law

Why: Comparing the Sine Law and Cosine Law requires students to have prior knowledge of the conditions and applications of the Sine Law.

Key Vocabulary

Cosine Law	A mathematical formula relating the lengths of the sides of a triangle to the cosine of one of its angles. It is used for non-right triangles when two sides and the included angle (SAS) or all three sides (SSS) are known.
Included Angle	The angle formed by two sides of a triangle. In the context of the Cosine Law, knowing two sides and the included angle is a condition for its application.
Solve a Triangle	To find the measures of all unknown sides and angles of a triangle. This involves using trigonometric laws like the Sine Law or Cosine Law.
Ambiguous Case	A situation in trigonometry, typically associated with the Sine Law, where given information can lead to two possible triangles. The Cosine Law, under its standard conditions (SAS or SSS), avoids this ambiguity.

Watch Out for These Misconceptions

Common Misconception

Cosine Law applies only to acute triangles.

What to Teach Instead

The law works for obtuse angles too, where cosine is negative, making the term positive and fitting the geometry. Physical models built by students reveal this, as they measure obtuse triangles and compute, comparing to acute cases in pairs.

Common Misconception

Cosine Law requires the same givens as Pythagoras.

What to Teach Instead

Pythagoras is a special case for right triangles; Cosine Law needs SAS or SSS. Sorting activities with triangle cards help students classify givens and select the right law through group consensus.

Common Misconception

Always use Cosine Law when an angle is known.

What to Teach Instead

Sine Law suits AAS or SSA; Cosine fits SAS or SSS. Relay races practicing selection build quick recognition, with peer review catching overgeneralizations.

Active Learning Ideas

Pairs Practice: Build and Verify

Partners receive cards with two sides and included angle. One builds the triangle using rulers and protractors, measures the third side, and calculates with Cosine Law. They switch roles and compare results, discussing discrepancies.

25 min·Pairs

Small Groups: Law Selection Relay

Divide class into teams. Each member solves a triangle problem by choosing Sine or Cosine Law, writes the answer, and tags the next teammate. First team correct wins. Debrief choices as a class.

35 min·Small Groups

Whole Class: Problem Design Challenge

Pose real-world contexts like navigation or architecture. Students in rows brainstorm and design a Cosine Law problem, then pass to adjacent row for solving. Circulate to facilitate sharing.

40 min·Whole Class

Individual: Error Hunt Stations

Set up stations with solved triangles containing common errors. Students identify mistakes in law application, correct them, and explain in journals why the fix works.

30 min·Individual

Real-World Connections

• Surveyors use the Cosine Law to calculate distances and boundaries when direct measurement is impossible, such as determining the length of a property line that is obstructed by a building or a natural feature.
• Navigators on ships or aircraft use the Cosine Law to determine distances and bearings between points when only certain distances and angles are known, especially when dealing with non-right-angled paths.
• Physicists apply the Cosine Law when analyzing forces acting on an object at an angle to each other, such as calculating the resultant force when two forces are applied to a point at a known angle.

Assessment Ideas

Quick Check

Present students with three triangle scenarios, each with different given information (e.g., ASA, AAS, SAS, SSS, SSA). Ask students to write down which trigonometric law (Sine or Cosine) would be the most efficient to use for each scenario and briefly explain why.

Exit Ticket

Provide students with a triangle diagram showing sides a=10, b=12, and angle C=50 degrees. Ask them to calculate the length of side c using the Cosine Law and show their steps. Then, ask them to write one sentence explaining why the Cosine Law was appropriate here.

Discussion Prompt

Pose the question: 'Imagine you are designing a navigation system. What information would you need to input to ensure the system can accurately calculate distances using the Cosine Law, and why is this information crucial?' Facilitate a class discussion on the requirements for applying the Cosine Law in a practical context.

Frequently Asked Questions

When should students use the Cosine Law instead of the Sine Law?

Use Cosine Law for SAS (two sides, included angle) or SSS (three sides) configurations. Sine Law fits when you have a pair including angle opposite a side, like SSA or AAS. Teach this through card sorts where students match givens to laws, then solve, reinforcing the distinction in context.

What are real-world applications of the Cosine Law?

Applications include surveying distances between points with known baselines and angles, like in land measurement or GPS triangulation. In physics, it calculates resultant forces with known magnitudes and angle between them. Engineering uses it for truss analysis. Have students design problems from news articles on these fields to connect math to careers.

How does active learning help teach the Cosine Law?

Active approaches like building triangles with materials let students derive the law empirically by measuring and comparing to formulas. Collaborative relays for law selection practice quick decisions under time pressure. Designing problems fosters ownership, as groups defend choices in presentations, deepening understanding beyond rote memorization.

How can I address common errors with the Cosine Law?

Errors often stem from sign mishandling in obtuse cases or confusing configurations. Use error-hunt stations where students spot and fix mistakes, then share in small groups. Follow with guided practice mixing laws, providing immediate feedback to build accuracy and confidence.