Class: Senior Secondary School 2 (SS2 / SSS2)
Term: Third Term
Week: 3
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Trigonometry
Previous Lesson: Trigonometric Function: Graph Relationships and Inverse Graphs.
Topic: TRIGONOMETRIC FUNCTION
Subject Matter: Solution of simple equation involving the six trigonometric functions, proofs of simple trigonometric identities (sin^2 x + cos^2 x = 1, sec^2 x = 1 + tan^2 x)

Specific Objectives

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

Cognitive Domain:

Define simple trigonometric equations and identities.
Solve simple equations involving sine, cosine, and tangent functions.
State the fundamental trigonometric identities.
Explain the steps involved in proving trigonometric identities.

Affective Domain:

Appreciate the importance of trigonometric identities in solving complex problems.
Show interest in solving trigonometric equations and proving identities.
Develop a positive attitude towards problem-solving in Further Mathematics.

Psychomotor Domain:

Apply appropriate formulas to solve simple trigonometric equations.
Demonstrate the step-by-step proof of sin²x + cos²x = 1.
Demonstrate the step-by-step proof of sec²x = 1 + tan²x.

Collaborate with peers to discuss and solve trigonometric problems.
Communicate their understanding of trigonometric concepts clearly to others.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
New Further Mathematics Project 2 for Senior Secondary Schools.
https://www.mathsisfun.com/algebra/trig-equations.html
https://www.dummies.com/education/math/trigonometry/how-to-prove-trigonometric-identities/

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts showing sketches of inverse of sin x, cos x and tan x.
Whiteboard and markers.
Scientific calculators.
Textbooks.

Rationale for the Lesson

This lesson helps pupils understand how to find unknown angles in trigonometric expressions and verify fundamental relationships between trigonometric functions. This knowledge is important for solving problems in higher mathematics, physics, and engineering.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of basic trigonometric ratios (SOH CAH TOA), Pythagoras theorem, and properties of special angles (30°, 45°, 60°).

Lesson Content/Board Summary

TRIGONOMETRIC FUNCTIONS

Solving Simple Trigonometric Equations

A trigonometric equation is an equation that involves one or more trigonometric functions of a variable angle. Solving a trigonometric equation means finding the value(s) of the variable angle that satisfy the equation.

Steps to solve simple trigonometric equations:

Isolate the trigonometric function on one side of the equation.
Determine the reference angle using the inverse trigonometric function.
Identify the quadrants where the trigonometric function has the required sign.
Find all possible solutions within the specified range (e.g., 0° to 360°).

Example 1: Solve the equation sin x = 0.5 for 0° ≤ x ≤ 360°.

Solution:
Step 1: The trigonometric function is already isolated: sin x = 0.5.
Step 2: Find the reference angle. sin⁻¹(0.5) = 30°.
Step 3: Sine is positive in the 1st and 2nd quadrants.
Step 4: In the 1st quadrant, x = 30°.
In the 2nd quadrant, x = 180° – 30° = 150°.

Therefore, x = 30° or 150°.

Example 2: Solve the equation 2cos x + 1 = 0 for 0° ≤ x ≤ 360°.

Solution:
Step 1: Isolate the trigonometric function:
2cos x = -1
cos x = -1/2
Step 2: Find the reference angle. cos⁻¹(1/2) = 60°.
Step 3: Cosine is negative in the 2nd and 3rd quadrants.
Step 4: In the 2nd quadrant, x = 180° – 60° = 120°.
In the 3rd quadrant, x = 180° + 60° = 240°.

Therefore, x = 120° or 240°.

Proofs of Simple Trigonometric Identities

A trigonometric identity is an equation that is true for all values of the variable for which both sides of the equation are defined. Identities are used to simplify expressions and solve equations.

Fundamental Identities:

1. Proof of sin²x + cos²x = 1

Consider a right-angled triangle with angle x, opposite side ‘o’, adjacent side ‘a’, and hypotenuse ‘h’.
From Pythagoras theorem, o² + a² = h² (Equation 1)

We know that:
sin x = o/h => sin²x = o²/h²
cos x = a/h => cos²x = a²/h²

Adding sin²x and cos²x:
sin²x + cos²x = o²/h² + a²/h²
sin²x + cos²x = (o² + a²)/h²

From Equation 1, substitute o² + a² = h²:
sin²x + cos²x = h²/h²
sin²x + cos²x = 1

2. Proof of sec²x = 1 + tan²x

We start from the identity: sin²x + cos²x = 1

Divide all terms by cos²x (assuming cos x ≠ 0):
(sin²x / cos²x) + (cos²x / cos²x) = 1 / cos²x

We know that:
sin x / cos x = tan x => sin²x / cos²x = tan²x
1 / cos x = sec x => 1 / cos²x = sec²x

Substitute these into the equation:
tan²x + 1 = sec²x
sec²x = 1 + tan²x

Note: Another important identity derived similarly is cosec²x = 1 + cot²x (by dividing sin²x + cos²x = 1 by sin²x).

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 their previous knowledge of basic trigonometric ratios (SOH CAH TOA) and special angles, asking questions like “What is sin 30°?”
Pupils’ Activity: Pupils respond to the teacher’s questions and recall prior knowledge of trigonometry.
Learning Point: Pupils recall and connect previous knowledge to the new topic.

Step 2: Presentation of Simple Trigonometric Equations

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher defines what a simple trigonometric equation is and explains the general steps for solving them, using the whiteboard to illustrate. The teacher works through Example 1 (sin x = 0.5) from the board summary.
Pupils’ Activity: Pupils listen attentively, copy notes, and ask questions for clarification.
Learning Point: Pupils understand the concept of trigonometric equations and the steps to solve them.

Step 3: Solving Examples of Simple Trigonometric Equations

Time: 8 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher guides pupils through solving Example 2 (2cos x + 1 = 0) on the whiteboard, encouraging pupils to suggest steps.
Pupils’ Activity: Pupils actively participate, suggest steps, and solve the example in their notebooks.
Learning Point: Pupils practice solving trigonometric equations with teacher guidance.

Step 4: Introduction to Trigonometric Identities

Time: 5 minutes
Teaching Skill: Definition/Explanation
Teacher’s Activity: The teacher introduces trigonometric identities, explaining that they are equations true for all valid values of the variable, and differentiates them from conditional equations.
Pupils’ Activity: Pupils listen and understand the definition and purpose of trigonometric identities.
Learning Point: Pupils grasp the meaning and significance of trigonometric identities.

Step 5: Proof of sin²x + cos²x = 1

Time: 7 minutes
Teaching Skill: Demonstration/Explanation
Teacher’s Activity: The teacher demonstrates the proof of the identity sin²x + cos²x = 1 on the whiteboard, using a right-angled triangle and Pythagoras theorem, explaining each step clearly.
Pupils’ Activity: Pupils observe the proof, ask questions, and copy the proof into their notes.
Learning Point: Pupils learn how to derive the fundamental trigonometric identity.

Step 6: Proof of sec²x = 1 + tan²x

Time: 5 minutes
Teaching Skill: Demonstration/Derivation
Teacher’s Activity: The teacher guides pupils through the derivation of sec²x = 1 + tan²x from the identity sin²x + cos²x = 1, explaining the division by cos²x.
Pupils’ Activity: Pupils follow the derivation, contribute ideas, and record the proof in their notebooks.
Learning Point: Pupils understand how to derive other identities from the fundamental one.

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 a trigonometric identity.
Solve for x in the equation tan x = 1 for 0° ≤ x ≤ 360°.
Prove the identity sin²θ + cos²θ = 1.
State another fundamental trigonometric identity derived from sin²θ + cos²θ = 1.

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 learning points, emphasizing the importance of understanding trigonometric equations and identities for future mathematical studies.
Pupils’ Activity: Pupils listen to the summary and prepare for the next lesson.
Learning Point: Pupils consolidate their understanding of the topic.

Lesson Keywords

Trigonometric function – A function of an angle (sine, cosine, tangent, etc.) used to relate the angles of a triangle to the lengths of its sides.
Equation – A statement that two mathematical expressions are equal.
Identity – An equation that is true for all values of the variables involved.
Sine (sin) – The ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
Cosine (cos) – The ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.
Tangent (tan) – The ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle.
Secant (sec) – The reciprocal of the cosine function (1/cos x).
Cosecant (cosec) – The reciprocal of the sine function (1/sin x).
Cotangent (cot) – The reciprocal of the tangent function (1/tan x).

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex trigonometric equations involving double angles or quadratic forms. For pupils who need more support, the teacher can offer additional guided practice with simpler equations and step-by-step hints for proofs, using visual aids and one-on-one assistance.

Note for teachers using this lesson plan

Ensure pupils have a solid foundation in basic algebra and trigonometry before introducing this topic. Emphasize the graphical interpretation of solutions for trigonometric equations to provide a deeper understanding. Encourage pupils to practice proving identities regularly, as this builds problem-solving skills and reinforces fundamental concepts. Provide ample opportunities for pupils to ask questions and work through problems collaboratively.