Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: 1st Term
Week: 9
Age: 15 years
Duration: 45 minutes
Subject: Further Maths
Curriculum Theme: Further Mathematics
Previous Lesson: Logical Reasoning I.
Topic: TRIGONOMETRIC RATIOS OF SPECIAL ANGLES
Subject Matter: trigonometric ratios of 30 degrees 45 degrees and 60 degrees, application of special angle ratios without tables

Specific Objectives

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

Cognitive Domain:

State the trigonometric ratios for 30°, 45°, and 60°.
Derive the trigonometric ratios for special angles (30°, 45°, 60°).
Calculate exact values of trigonometric expressions involving special angles without using tables or calculators.

Affective Domain:

Appreciate the importance of special angles in solving mathematical problems.
Show interest in deriving and applying trigonometric ratios.

Psychomotor Domain:

Draw appropriate right-angled triangles to aid the derivation of ratios.
Solve problems involving trigonometric ratios of special angles.

Work cooperatively in groups to derive and discuss ratios.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum (Senior Secondary Further Mathematics)
State Unified Scheme of Work (Further Mathematics SSS 1)
New Further Mathematics for Senior Secondary Schools by P.N. Okeke
New General Mathematics for Senior Secondary Schools Book 1

Instructional Materials

The teacher will teach this lesson with the aid of:

Right-angled triangle diagrams (e.g., equilateral triangle, isosceles right-angled triangle)
Plane figures drawn on the board
Whiteboard/chalkboard
Markers/chalk

Rationale for the Lesson

This lesson helps pupils understand the exact trigonometric values for common angles without relying on calculators. Knowing these special ratios is important for simplifying expressions and solving problems in higher mathematics and various real-world applications involving angles and distances.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of basic trigonometric ratios (sine, cosine, tangent) and the properties of special triangles such as equilateral triangles and isosceles right-angled triangles.

Lesson Content/Board Summary

TRIGONOMETRIC RATIOS OF SPECIAL ANGLES

Introduction to Special Angles

Special angles in trigonometry are angles whose trigonometric ratios (sine, cosine, tangent) can be determined exactly without the use of a calculator or trigonometric tables. The most common special angles are 30°, 45°, and 60°.

Derivation of Ratios for 30° and 60°

Consider an equilateral triangle ABC with side length 2 units. All angles are 60°.

Draw an altitude AD from A to BC. AD bisects BC and angle BAC.

In right-angled triangle ABD:

Angle B = 60°
Angle BAD = 30°
Hypotenuse AB = 2
Adjacent side BD = 1
Opposite side AD = sqrt(AB² – BD²) = sqrt(2² – 1²) = sqrt(3)

From triangle ABD:

For 30°:

sin 30° = Opposite/Hypotenuse = BD/AB = 1/2
cos 30° = Adjacent/Hypotenuse = AD/AB = sqrt(3)/2
tan 30° = Opposite/Adjacent = BD/AD = 1/sqrt(3) = sqrt(3)/3

For 60°:

sin 60° = Opposite/Hypotenuse = AD/AB = sqrt(3)/2
cos 60° = Adjacent/Hypotenuse = BD/AB = 1/2
tan 60° = Opposite/Adjacent = AD/BD = sqrt(3)/1 = sqrt(3)

Derivation of Ratios for 45°

Consider an isosceles right-angled triangle PQR, with PQ = QR = 1 unit. The angles are 45°, 45°, and 90°.

By Pythagoras’ theorem, PR² = PQ² + QR² = 1² + 1² = 2, so PR = sqrt(2).

In right-angled triangle PQR (taking angle P or R as 45°):

Angle P = 45°
Opposite side QR = 1
Adjacent side PQ = 1
Hypotenuse PR = sqrt(2)

From triangle PQR:

For 45°:

sin 45° = Opposite/Hypotenuse = QR/PR = 1/sqrt(2) = sqrt(2)/2
cos 45° = Adjacent/Hypotenuse = PQ/PR = 1/sqrt(2) = sqrt(2)/2
tan 45° = Opposite/Adjacent = QR/PQ = 1/1 = 1

Summary Table of Special Angle Ratios

The trigonometric ratios for 30°, 45°, and 60° are summarized below:

Angle (θ) | sin θ | cos θ | tan θ
30° | 1/2 | sqrt(3)/2 | 1/sqrt(3) or sqrt(3)/3
45° | 1/sqrt(2) or sqrt(2)/2 | 1/sqrt(2) or sqrt(2)/2 | 1
60° | sqrt(3)/2 | 1/2 | sqrt(3)

Application of Special Angle Ratios

These ratios are used to evaluate expressions or solve problems without using calculators.

Examples:

Evaluate sin 60° + cos 30°.
Simplify (tan 45° + sin 30°) / cos 60°.
Find the exact value of 2 sin 30° cos 60°.

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 asks them to recall the basic trigonometric ratios (SOH CAH TOA) and how they relate to the sides of a right-angled triangle. The teacher might ask for examples of angles whose ratios they already know (e.g., from tables if they’ve used them).
Pupils’ Activity: Pupils respond to the teacher’s questions and recall prior knowledge of trigonometric ratios.
Learning Point: Pupils are reminded of basic trigonometric concepts and prepared for the lesson.

Step 2: Presentation of Topic

Time: 2 minutes
Teaching Skill: Introduction
Teacher’s Activity: The teacher introduces the topic “Trigonometric Ratios of Special Angles” and explains that today’s lesson will focus on deriving and applying the ratios for 30°, 45°, and 60° without using calculators.
Pupils’ Activity: Pupils listen attentively and write down the topic in their notebooks.
Learning Point: Pupils know the specific focus of the lesson.

Step 3: Derivation of Ratios for 30° and 60°

Time: 10 minutes
Teaching Skill: Demonstration/Explanation
Teacher’s Activity: The teacher draws an equilateral triangle on the board and guides pupils to divide it into two right-angled triangles by drawing an altitude. The teacher then leads pupils through the process of determining the side lengths and applying SOH CAH TOA to derive the sine, cosine, and tangent for 30° and 60°. The teacher writes the derived ratios on the board.
Pupils’ Activity: Pupils observe the derivation, participate in determining side lengths, and copy the derivations and ratios into their notebooks.
Learning Point: Pupils understand how the ratios for 30° and 60° are obtained.

Step 4: Derivation of Ratios for 45°

Time: 8 minutes
Teaching Skill: Demonstration/Guidance
Teacher’s Activity: The teacher draws an isosceles right-angled triangle on the board and guides pupils to determine its side lengths using Pythagoras’ theorem. The teacher then leads pupils to apply SOH CAH TOA to derive the sine, cosine, and tangent for 45°. The teacher writes the derived ratios on the board.
Pupils’ Activity: Pupils observe, participate in calculations, and copy the derivations and ratios into their notebooks.
Learning Point: Pupils understand how the ratios for 45° are obtained.

Step 5: Summary and Memorization

Time: 5 minutes
Teaching Skill: Consolidation
Teacher’s Activity: The teacher compiles all the derived special angle ratios (30°, 45°, 60°) into a clear table on the board. The teacher encourages pupils to find patterns or mnemonics to help memorize these ratios.
Pupils’ Activity: Pupils copy the summary table and try to identify patterns for memorization.
Learning Point: Pupils have a consolidated reference for the special angle ratios.

Step 6: Application of Special Angle Ratios

Time: 10 minutes
Teaching Skill: Problem Solving/Demonstration
Teacher’s Activity: The teacher presents several examples from the “Application of Special Angle Ratios” section of the board summary. The teacher demonstrates how to substitute the exact values from the table to evaluate trigonometric expressions without using a calculator. The teacher gives pupils simple exercises to try.
Pupils’ Activity: Pupils pay attention to the examples, ask questions for clarification, and attempt the given exercises.
Learning Point: Pupils learn to apply the special angle ratios in calculations.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

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

Define what is meant by “special angles” in trigonometry.
Using an equilateral triangle, derive the value of sin 60°.
State the exact value of tan 45°.
Evaluate cos 30° + sin 30°.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the importance of knowing and applying the special angle ratios. The teacher assigns homework involving more application problems.
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: The lesson is concluded with a recap and practice opportunities.

Lesson Keywords

Trigonometric ratios – Relationships between the angles and sides of a right-angled triangle.
Special angles – Angles (30°, 45°, 60°) whose trigonometric ratios have exact values.
Sine (sin) – Opposite side / Hypotenuse.
Cosine (cos) – Adjacent side / Hypotenuse.
Tangent (tan) – Opposite side / Adjacent side.
Derivation – The process of obtaining a mathematical formula or value from basic principles.

Differentiation

For pupils who grasp the concept quickly, the teacher can provide more complex application problems or challenge them to derive ratios for related angles. For pupils who are struggling, the teacher will provide additional guided practice, simplify the derivation steps, and offer mnemonic devices to help memorize the ratios.

Note for teachers using this lesson plan

Teachers should ensure that pupils are comfortable with basic trigonometry before this lesson. Emphasize the visual aspect of the derivations using clear diagrams. Encourage pupils to memorize the summary table of ratios as it forms a foundation for future topics. Provide ample practice exercises for applying these ratios in various expressions.