Mathematics - Junior Secondary 3 - Trigonometry

Trigonometry

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TERM: 2ND TERM

WEEK 8
Class: Junior Secondary School 3
Age: 14 years
Duration: 40 minutes of 5 periods
Subject: Mathematics
Topic: Trigonometry
Focus: Sine, Cosine, and Tangent of Acute Angles

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

  1. Understand the trigonometric ratios: sine, cosine, and tangent of an acute angle.
  2. Identify and label the sides of a right-angled triangle.
  3. Calculate the sine, cosine, and tangent of an acute angle in a right-angled triangle.
  4. Solve real-life problems using trigonometric ratios.
  5. Apply trigonometry to find unknown angles and sides of right-angled triangles.

INSTRUCTIONAL TECHNIQUES:
• Question and answer
• Guided demonstration
• Discussion
• Drills and exercises
• Real-life application

INSTRUCTIONAL MATERIALS:
• Trigonometric charts
• Protractors
• Calculators
• Whiteboard and marker
• Right-angled triangle flashcards

PERIOD 1 & 2: Introduction to Trigonometric Ratios: Sine, Cosine, and Tangent

PRESENTATION:

Step

Teacher’s Activity

Pupil’s Activity

Step 1 - Introduction

Introduces trigonometric ratios (Sine, Cosine, and Tangent). Explains the relationship between the sides of a right-angled triangle.

Pupils listen, ask, and answer questions.

Step 2 - Explanation

Defines sine, cosine, and tangent using the right-angled triangle. Formulae: - Sine = Opposite / Hypotenuse - Cosine = Adjacent / Hypotenuse - Tangent = Opposite / Adjacent

Pupils repeat the formulae and understand the definitions.

Step 3 - Demonstration

Draws a right-angled triangle on the board. Identifies the hypotenuse, opposite, and adjacent sides in relation to a given angle.

Pupils observe and identify the sides of the triangle.

Step 4 - Note Taking

Pupils take notes of the trigonometric ratios and the sides of a right-angled triangle.

Pupils copy notes.

NOTE ON BOARD:

  • Sine (θ) = Opposite / Hypotenuse
  • Cosine (θ) = Adjacent / Hypotenuse
  • Tangent (θ) = Opposite / Adjacent

EVALUATION (5 exercises):

  1. Label the sides of a right-angled triangle given the angle θ.
  2. Find the sine of a right-angled triangle with an opposite side of 4 and a hypotenuse of 5.
  3. Find the cosine of a right-angled triangle with an adjacent side of 3 and a hypotenuse of 5.
  4. Find the tangent of a right-angled triangle with an opposite side of 6 and an adjacent side of 8.
  5. Identify the ratio of sine, cosine, and tangent in the triangle with sides: 3, 4, and 5.

CLASSWORK (5 questions):

  1. Find the sine of an angle in a triangle with opposite = 7, hypotenuse = 10.
  2. Find the cosine of an angle in a triangle with adjacent = 8, hypotenuse = 10.
  3. Find the tangent of an angle in a triangle with opposite = 5, adjacent = 12.
  4. Calculate the value of tan(30°) using a calculator.
  5. Label the sides of a triangle with sides: 5, 12, 13.

ASSIGNMENT (5 tasks):

  1. Solve for the sine, cosine, and tangent for angles in a right-angled triangle where:
    • Opposite = 6, Hypotenuse = 10
    • Adjacent = 8, Hypotenuse = 17
  2. Draw a right-angled triangle and label the sine, cosine, and tangent for a 45° angle.
  3. Find the hypotenuse of a triangle with an opposite side of 7 and a sine of 0.7.
  4. Solve for the adjacent side of a triangle with an angle of 60° and a hypotenuse of 20.
  5. Calculate the sine, cosine, and tangent of a 90° angle.

PERIOD 3 & 4: Applications of Trigonometric Ratios to Solve Problems on Angles and Sides of a Right-Angled Triangle

PRESENTATION:

Step

Teacher’s Activity

Pupil’s Activity

Step 1 - Introduction

Revises trigonometric ratios and introduces their application in solving for unknown angles and sides.

Pupils listen and answer review questions.

Step 2 - Explanation

Solves problems using trigonometric ratios to find missing sides or angles in right-angled triangles. Shows the steps:

1.   Identify the sides.

2.   Apply the correct trigonometric ratio.

3.   Solve for the unknown.

Pupils follow the steps and solve a sample problem.

Step 3 – Demonstration

Solves a word problem involving the application of sine, cosine, or tangent. Example: A ladder leans against a wall. Find the height it reaches on the wall using the angle and length of the ladder.

Pupils practice solving similar problems

Step 4 - Practice

Pupils solve practice problems independently or in pairs with teacher’s assistance.

Pupils work individually or in groups

 

NOTE ON BOARD:
Example problem:
A 10-meter ladder leans against a wall. The angle between the ladder and the ground is 30°. How high up the wall does the ladder reach?
Solution:

  • Use the sine formula:
    • Sine(30°) = Opposite / Hypotenuse
    • 0.5 = x / 10
    • x = 5 meters (height the ladder reaches on the wall).

EVALUATION (5 exercises):

  1. A 15-meter ladder leans against a wall at an angle of 60°. How high does the ladder reach?
  2. A right-angled triangle has a hypotenuse of 13 meters and an adjacent side of 12 meters. Find the angle θ using cosine.
  3. Find the height of a building if the angle of elevation from a point 50 meters away from the base is 45°.
  4. In a triangle, the opposite side is 8 meters and the hypotenuse is 10 meters. Find the angle.
  5. A rope of length 20 meters is tied to a pole. The angle between the ground and the rope is 45°. How high is the pole?

CLASSWORK (5 questions):

  1. A triangle has a hypotenuse of 20 meters and an opposite side of 16 meters. Find the sine of the angle.
  2. A triangle has an adjacent side of 9 meters and an opposite side of 12 meters. Find the tangent of the angle.
  3. Calculate the angle θ in a triangle with a hypotenuse of 5 meters and an opposite side of 3 meters.
  4. A right-angled triangle has a hypotenuse of 10 meters and an angle of 45°. Find the opposite side.
  5. Find the height of a building using the tangent ratio if the distance from the building is 30 meters and the angle of elevation is 60°.

ASSIGNMENT (5 tasks):

  1. A right-angled triangle has a hypotenuse of 25 meters and an angle of 30°. Find the opposite and adjacent sides.
  2. Solve for the angle in a triangle with an opposite side of 9 meters and an adjacent side of 12 meters.
  3. A right-angled triangle has an adjacent side of 6 meters and an opposite side of 8 meters. Find the tangent of the angle.
  4. Calculate the height of a building if the distance from the point of observation is 50 meters and the angle of elevation is 36°.
  5. Solve for the missing angle in a triangle with sides: opposite = 4 meters, adjacent = 3 meters.

 

PERIOD 5: Real-Life Application of Trigonometry in Word Problems

PRESENTATION:

Step

Teacher’s Activity

Pupil’s Activity

Step 1 - Introduction

Introduces real-life applications of trigonometry in fields like architecture, engineering, and navigation.

Pupils listen and participate in discussions.

Step 2 - Examples

Presents a word problem about a tower, where trigonometric ratios are used to calculate the height.

Pupils identify and solve similar word problems.

Step 3 - Drill

Pupils solve real-life word problems involving the use of trigonometry to find angles and sides of triangles.

Pupils work in pairs and discuss their solutions.

Step 4 - Practice

Teacher assigns word problems for independent practice.

Pupils complete the problems and submit for review.

EVALUATION (5 questions):

  1. A man standing 20 meters away from a building sees the top at an angle of 45°. Find the height of the building.
  2. A person is standing 40 meters from a tall pole. The angle of elevation to the top of the pole is 30°. Find the height of the pole.
  3. The angle of depression from the top of a cliff to a boat on the water is 25°. If the boat is 100 meters away from the cliff, find the height of the cliff.
  4. A right-angled triangle has a hypotenuse of 20 meters and an angle

of 60°. Find the adjacent and opposite sides.
5. A man climbs a 15-meter ladder leaning against a wall. The angle between the ladder and the ground is 40°. How high does the ladder reach on the wall?

CLASSWORK (5 questions):

  1. A tree casts a shadow 12 meters long when the angle of elevation of the sun is 30°. Find the height of the tree.
  2. A building stands 50 meters tall. The angle of elevation from a point 50 meters away from the base is 40°. Calculate the height using trigonometric ratios.
  3. A 25-meter-long cable is tied to a flagpole at an angle of 45°. Find the height of the flagpole.
  4. In a triangle, the opposite side is 10 meters, and the hypotenuse is 20 meters. Find the sine of the angle.

Calculate the angle of elevation if the opposite side is 8 meters, and the adjacent side is 6 meters.