Lesson Note on Trigonometry: Sine Rule and Cosine Rule for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: 2nd Term
Week: 5 – 6
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Geometry and Mensuration
Previous Lesson: Algebraic Fractions: Equations and Substitution.
Topic: TRIGONOMETRY (I)
Subject Matter: Derivation of sine rule and its application, derivation of cosine rule and its applications

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

• State the Sine Rule and the Cosine Rule.
• Derive the formula for both the Sine Rule and the Cosine Rule.
• Apply the Sine Rule to solve problems involving triangles.
• Apply the Cosine Rule to solve problems involving triangles.

Affective Domain:

• Appreciate the importance of trigonometric rules in solving real-world problems.
• Participate actively in the derivation and problem-solving processes.

Psychomotor Domain:

• Draw and label triangles accurately to represent given problems.
• Correctly substitute values into the Sine and Cosine Rule formulas to find unknown sides or angles.

Social Domain:

• Collaborate with peers to discuss different approaches to solving trigonometric problems.

Reference Materials

The following resources were used in planning this lesson:

• 9 Years Basic Education Curriculum
• State Unified Scheme of Work
• New General Mathematics for Senior Secondary Schools 2
• https://www.mathsisfun.com/algebra/trig-sine-law.html
• https://www.mathsisfun.com/algebra/trig-cosine-law.html

Instructional Materials

The teacher will teach this lesson with the aid of:

• Acute angle chart
• Obtuse angled triangle chart
• Whiteboard and markers
• Textbooks
• Calculators

Rationale for the Lesson

This lesson helps pupils understand how to solve for unknown sides and angles in non-right-angled triangles. This knowledge is important for various applications in engineering, surveying, navigation, and other fields where indirect measurements are often needed.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of basic trigonometry (SOH CAH TOA), Pythagoras theorem, and properties of triangles, including acute and obtuse angles.

Lesson Content/Board Summary

TRIGONOMETRY (I)

Derivation of Sine Rule and its Applications

The Sine Rule is a relationship between the sides and angles of any triangle. It states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.

Formula: For any triangle ABC with sides a, b, c opposite angles A, B, C respectively:

a / sin A = b / sin B = c / sin C

When to use the Sine Rule:

• When you know two angles and one side (AAS or ASA).
• When you know two sides and a non-included angle (SSA).

Derivation of the Sine Rule:

Consider a triangle ABC. Draw a perpendicular ‘h’ from vertex C to side AB. Let the foot of the perpendicular be D.

In right-angled triangle ADC:

sin A = h / b => h = b sin A ……(1)

In right-angled triangle BDC:

sin B = h / a => h = a sin B ……(2)

Equating (1) and (2):

b sin A = a sin B

Dividing both sides by (sin A sin B):

b / sin B = a / sin A

Similarly, by drawing a perpendicular from A to BC, we can show that a / sin A = c / sin C. Thus, the Sine Rule is derived.

Applications/Examples of Sine Rule:

Example 1: Finding a side

In triangle PQR, angle P = 60°, angle Q = 45°, and side r = 10 cm. Find side p.

Solution:

First, find angle R:

Angle R = 180° – (60° + 45°) = 180° – 105° = 75°

Using the Sine Rule: p / sin P = r / sin R

p / sin 60° = 10 / sin 75°

p = (10 * sin 60°) / sin 75°

p = (10 * 0.8660) / 0.9659

p = 8.660 / 0.9659

p ≈ 8.97 cm (to 3 significant figures)

Example 2: Finding an angle

In triangle XYZ, x = 12 cm, y = 15 cm, and angle X = 40°. Find angle Y.

Solution:

Using the Sine Rule: x / sin X = y / sin Y

12 / sin 40° = 15 / sin Y

sin Y = (15 * sin 40°) / 12

sin Y = (15 * 0.6428) / 12

sin Y = 9.642 / 12

sin Y = 0.8035

Y = sin⁻¹(0.8035)

Y ≈ 53.47° (to 2 decimal places)

NOTE: Remember to consider the possibility of an obtuse angle for SSA cases, as sine is positive in both the first and second quadrants.

Derivation of Cosine Rule and its Applications

The Cosine Rule relates the lengths of the sides of a triangle to the cosine of one of its angles. It is a generalization of the Pythagorean theorem.

Formula: For any triangle ABC with sides a, b, c opposite angles A, B, C respectively:

• a² = b² + c² – 2bc cos A
• b² = a² + c² – 2ac cos B
• c² = a² + b² – 2ab cos C

When to use the Cosine Rule:

• When you know two sides and the included angle (SAS).
• When you know all three sides (SSS).

Derivation of the Cosine Rule (for side ‘a’):

Consider a triangle ABC. Draw a perpendicular ‘h’ from vertex C to side AB. Let the foot of the perpendicular be D. Let AD = x, so DB = c – x.

In right-angled triangle ADC:

h² = b² – x² ……(1) (by Pythagoras theorem)

Also, cos A = x / b => x = b cos A ……(2)

In right-angled triangle BDC:

a² = h² + (c – x)² ……(3) (by Pythagoras theorem)

Substitute (1) into (3):

a² = (b² – x²) + (c – x)²

a² = b² – x² + c² – 2cx + x²

a² = b² + c² – 2cx

Now substitute (2) into this equation:

a² = b² + c² – 2c(b cos A)

a² = b² + c² – 2bc cos A

Similarly, the other forms of the Cosine Rule can be derived.

Applications/Examples of Cosine Rule:

Example 1: Finding a side (SAS)

In triangle MNO, m = 8 cm, n = 10 cm, and angle O = 60°. Find side o.

Solution:

Using the Cosine Rule: o² = m² + n² – 2mn cos O

o² = 8² + 10² – 2 * 8 * 10 * cos 60°

o² = 64 + 100 – 160 * 0.5

o² = 164 – 80

o² = 84

o = √84

o ≈ 9.17 cm (to 3 significant figures)

Example 2: Finding an angle (SSS)

In triangle ABC, a = 7 cm, b = 5 cm, and c = 8 cm. Find angle B.

Solution:

Using the Cosine Rule (rearranged to find an angle):

b² = a² + c² – 2ac cos B

2ac cos B = a² + c² – b²

cos B = (a² + c² – b²) / (2ac)

cos B = (7² + 8² – 5²) / (2 * 7 * 8)

cos B = (49 + 64 – 25) / 112

cos B = (113 – 25) / 112

cos B = 88 / 112

cos B = 0.7857

B = cos⁻¹(0.7857)

B ≈ 38.21° (to 2 decimal places)

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher greets the pupils and reviews the previous lesson on basic trigonometric ratios (SOH CAH TOA) and their application to right-angled triangles. The teacher then introduces the day’s topic by asking if these ratios can be used for non-right-angled triangles.
Pupils’ Activity: Pupils respond to greetings, recall previous knowledge, and engage in the discussion.
Learning Point: Pupils recall prior knowledge and are prepared for the new topic.

Step 2: Introduction to Sine Rule and its Derivation

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher displays the chart of acute and obtuse angles and guides pupils on how to denote vertices and corresponding sides of triangles. The teacher then introduces the Sine Rule and guides pupils through its derivation using a clear diagram on the board.
Pupils’ Activity: Pupils study the charts, follow the teacher’s guidance on notation, and observe the steps of the derivation.
Learning Point: Pupils understand the formula of the Sine Rule and its origin.

Step 3: Application of Sine Rule

Time: 10 minutes
Teaching Skill: Problem Solving/Demonstration
Teacher’s Activity: The teacher presents various examples illustrating the application of the Sine Rule to find unknown sides or angles in triangles, emphasizing when to use the rule. The teacher solves two examples step-by-step on the board.
Pupils’ Activity: Pupils pay attention to the examples, ask questions, and copy the solutions.
Learning Point: Pupils can apply the Sine Rule to solve problems.

Step 4: Introduction to Cosine Rule and its Derivation

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher introduces the Cosine Rule, explaining its purpose as a generalization of Pythagoras theorem. The teacher then guides pupils through the derivation of the Cosine Rule using a diagram.
Pupils’ Activity: Pupils listen, observe the derivation, and ask clarifying questions.
Learning Point: Pupils understand the formula of the Cosine Rule and how it is derived.

Step 5: Application of Cosine Rule

Time: 5 minutes
Teaching Skill: Problem Solving/Demonstration
Teacher’s Activity: The teacher provides examples demonstrating the application of the Cosine Rule to find unknown sides or angles, explaining when this rule is appropriate. The teacher solves examples on the board.
Pupils’ Activity: Pupils observe the problem-solving steps and note down the examples.
Learning Point: Pupils can apply the Cosine Rule to solve problems.

Step 6: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. State the Sine Rule for a triangle ABC.
2. State the Cosine Rule for finding side ‘a’ in a triangle ABC.
3. Mention two scenarios where the Sine Rule is typically used.
4. Mention two scenarios where the Cosine Rule is typically used.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 7: Conclusion

Time: 5 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the Sine and Cosine Rules, their derivations, and their applications. The teacher gives assignments for practice.
Pupils’ Activity: Pupils listen to the summary, ask any final questions, and copy the assignment.
Learning Point: Pupils consolidate their understanding and are given tasks to reinforce learning.

Lesson Keywords

• Sine Rule – A rule relating the ratio of a side length to the sine of its opposite angle in any triangle.
• Cosine Rule – A rule relating the lengths of the sides of a triangle to the cosine of one of its angles.
• Derivation – The process of showing how a formula or rule is obtained from fundamental principles.
• Acute angle – An angle less than 90 degrees.
• Obtuse angle – An angle greater than 90 degrees but less than 180 degrees.

Differentiation

For pupils who grasp concepts quickly, the teacher will provide more complex problems involving combined applications of both rules or real-world problem-solving scenarios. For pupils needing additional support, the teacher will provide simplified examples and offer one-on-one guidance during practice sessions, focusing on understanding the conditions for applying each rule.

Note for teachers using this lesson plan

Ensure pupils have a good understanding of basic trigonometry before this lesson. Emphasize the conditions for using each rule (SAS, SSS for Cosine Rule; AAS, ASA, SSA for Sine Rule). Encourage the use of calculators and guide pupils on how to use them effectively for trigonometric functions. Provide sufficient practice problems for both derivation and application.