Trigonometry (II)

Grade 10 · Mathematics

Semester 2 | Period 6 | Week 33

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Subject: Mathematics

Semester: 2

Period: 6

Week: 33

Week 33

Class: Grade 10
Age: 15 years
Duration: 40 minutes of 5 periods
Subject: Mathematics
Topic: Trigonometry (II)

SPECIFIC OBJECTIVES:

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

  1. Use sine, cosine, and tangent to solve problems involving right-angled triangles.
  2. Solve problems involving angles of elevation and depression using trigonometric ratios.
  3. Derive trigonometric ratios of 30°, 45°, and 60° without a calculator.
  4. Apply trigonometric ratios in solving real-life problems.

 

INSTRUCTIONAL TECHNIQUES:

  • Guided discovery
  • Question and answer
  • Demonstration
  • Group discussion
  • Practice exercises

 

INSTRUCTIONAL MATERIALS:

  • Charts showing unit triangles and trigonometric ratios
  • Mathematical instruments (ruler, protractor, compass)
  • Whiteboard and markers
  • Worksheet with exercises
  • Chalkboard and chalk

 

PERIOD 1 & 2: Introduction to Trigonometric Ratios and Their Applications

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Recaps basic trigonometric ratios: sine, cosine, and tangent. Defines them using a right-angled triangle.

Students listen and take notes.

Step 2 - Solving Right-Angled Triangle Problems

Demonstrates how to calculate unknown sides and angles in right-angled triangles using sine, cosine, and tangent.

Students observe and solve guided problems in class.

Step 3 - Angle of Elevation and Depression

Explains these concepts with diagrams and real-life examples (e.g., ladders, hills, buildings).

Students relate the concept to daily life and solve sample problems.

Step 4 - Practice Problems

Provides practice exercises involving right-angled triangles.

Students work individually or in pairs.

Step 5 - Feedback and Correction

Reviews the practice problems and corrects mistakes.

Students make corrections and ask questions.

 

NOTE ON BOARD:

  • sin θ = opposite/hypotenuse
  • cos θ = adjacent/hypotenuse
  • tan θ = opposite/adjacent
  • Angle of elevation: angle from horizontal up
  • Angle of depression: angle from horizontal down

 

EVALUATION (5 exercises):

  1. Solve for x in a right-angled triangle where sin θ = 3/5 and hypotenuse = 10 cm.
  2. Find the angle of elevation of the sun if a tree casts a shadow 10 m long and is 8 m tall.
  3. Use cosine to calculate the unknown side of a triangle where the adjacent = 5 cm and hypotenuse = 13 cm.
  4. Define angle of depression with a diagram.
  5. State the trigonometric ratio used when the opposite and adjacent sides are known.

 

CLASSWORK (5 questions):

  1. Use sine, cosine, and tangent to solve for missing sides in three different triangles.
  2. A ladder is leaning against a wall forming an angle of 60° with the ground. The ladder is 5 m long. How high up the wall does it reach?
  3. Identify when to use each of the trigonometric ratios.
  4. Draw and label a right-angled triangle with sides 3 cm, 4 cm, and 5 cm.
  5. Calculate an angle of depression using appropriate trigonometric ratio.

 

ASSIGNMENT (5 tasks):

  1. Find real-life examples where trigonometry is used (e.g., architecture, astronomy).
  2. Solve two problems involving angles of elevation.
  3. Construct a right-angled triangle and label its sides.
  4. Write short notes on the sine, cosine, and tangent functions.
  5. Use any triangle to demonstrate the use of tangent to find an unknown angle.

 

PERIOD 3 & 4: Deriving Trigonometric Ratios of 30°, 45°, and 60° Without Calculators

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Draws two special right triangles (isosceles and equilateral split in half).

Students observe and take notes.

Step 2 - 45° Triangle

Draws an isosceles right-angled triangle with equal sides of 1 unit and uses Pythagoras’ theorem to find the hypotenuse. Calculates sin, cos, and tan of 45°.

Students draw same and calculate ratios.

Step 3 - 30° and 60° Triangle

Constructs an equilateral triangle and splits it into two to derive 30° and 60° angles. Uses side ratios to calculate trigonometric values.

Students replicate and calculate.

Step 4 - Relationship and Patterns

Leads students to identify patterns in the ratios.

Students discuss and write their observations.

Step 5 - Practice

Provides guided practice with similar triangles.

Students work in pairs.

 

NOTE ON BOARD:

  • sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3
  • sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1
  • sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

 

EVALUATION (5 exercises):

  1. Derive the value of sin 30° from a triangle.
  2. Prove that tan 45° = 1 using triangle side ratios.
  3. Without using a calculator, find cos 60°.
  4. Draw a triangle to show sin 60°.
  5. Explain why sin 30° = cos 60°.

 

CLASSWORK (5 questions):

  1. Calculate sin, cos, and tan of 30°, 45°, and 60° from construction.
  2. Compare and contrast sin 30° and cos 60°.
  3. Explain why the tangent of 45° is always 1.
  4. Construct a triangle for 30° and find all three trigonometric ratios.
  5. Derive values of sine and cosine of 45° from the triangle.

 

ASSIGNMENT (5 tasks):

  1. Draw two special triangles for 30°-60° and 45°.
  2. Label the sides and derive the trigonometric ratios.
  3. Create a chart of sin, cos, and tan of 30°, 45°, and 60°.
  4. Write a short explanation of how to derive trigonometric values geometrically.
  5. Research other angles that can be derived without calculator and explain.

 

PERIOD 5: Conclusion and Review

PRESENTATION:

  • Summarizes key concepts: Trig ratios, angle of elevation/depression, 30°, 45°, and 60° derivations.
  • Revisits earlier problems and solves a few on the board.
  • Encourages questions from students and clarifies doubts.

 

EVALUATION:

  1. Oral quiz with rapid-response questions.
  2. Review and feedback on classwork and assignments.
  3. Students demonstrate how to derive a trigonometric ratio in front of class.