Application of trigonometric ratios of special angles to simple problems for SS 1

Class: SS 1
Term: First Term
Week: 6
Age: 15 years
Duration: 60 minutes
Subject: Mathematics
Curriculum Theme: Number and Numeration
Focal competence: Using trigonometric ratio knowledge to solve angles of elevation and depress 1on
Key competencies/values: Critical Thinking; Collaboration
Skills:

• Using trigonometric ratio knowledge to solve angles of elevation and depression

Previous Lesson: Number Base System and Conversion from any base to base 10
Topic: Trigonometry: Application Of Trigonometric Ratios Of Special Angles To Simple Problems
Subject Matter: Application of trigonometric ratios of special angles to simple problems, Application of trigonometry to solve angle of elevation and depression

Specific Objectives

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

Cognitive Domain

• Relate sine and cosine ratios to the unit circle.
• Explain the concept of angles of elevation and depression.
• Calculate unknown sides or angles in simple problems using special trigonometric ratios.

Psychomotor Domain

• Draw graphs of sine and cosine of angles.
• Sketch diagrams for problems involving angles of elevation and depression.
• Solve problems involving angles of elevation and depression using trigonometric ratios.

Affective Domain

• Appreciate the practical applications of trigonometry in real-life situations.
• Collaborate effectively in groups to solve trigonometric problems.

Reference Materials

The following resources were used in planning this lesson:

• 2025 New Revised Senior Secondary Education Curriculum (SSEC)
• Relevant State Unified Scheme of Work
• New General Mathematics for Senior Secondary Schools 1
• The HeadTeacher Scheme of work For The New Revised Senior Secondary Education Curriculum (SSEC)

Instructional Materials

The teacher will teach this lesson with the aid of:

• Chart showing trigonometric ratios of a right angled triangle
• Pencil and rulers
• Protractor
• Cut out shapes of right angled triangles showing angles 30° 45°, 60° and 90°
• Chart showing unit circle
• Whiteboard/Chalkboard
• Markers/Chalk

Rationale for the Lesson

This lesson is important as it provides students with practical skills to solve real-world problems using trigonometry. Understanding the application of special angle ratios and concepts like angles of elevation and depression is foundational for further studies in mathematics, physics, engineering, and surveying, enabling students to model and solve various scenarios.

Prerequisite/Previous Knowledge

Students should have prior knowledge of basic trigonometric ratios (sine, cosine, tangent), the properties of right-angled triangles, and the values of trigonometric ratios for special angles (30°, 45°, 60°).

Lesson Content/Board Summary

Trigonometry: Application Of Trigonometric Ratios Of Special Angles To Simple Problems

Review of Basic Trigonometric Ratios

In a right-angled triangle, the trigonometric ratios relate the angles to the lengths of the sides. For an acute angle (theta):

1. Sine ((sin theta)) = (frac{text{Opposite}}{text{Hypotenuse}})
2. Cosine ((cos theta)) = (frac{text{Adjacent}}{text{Hypotenuse}})
3. Tangent ((tan theta)) = (frac{text{Opposite}}{text{Adjacent}})

A common mnemonic to remember these ratios is SOH CAH TOA.

Trigonometric Ratios of Special Angles (30°, 45°, 60°)

These are angles whose trigonometric ratios can be determined without a calculator, often derived from equilateral triangles or squares.

Angle ((theta))	(sin theta)	(cos theta)	(tan theta)
30°	(frac{1}{2})	(frac{sqrt{3}}{2})	(frac{1}{sqrt{3}}) or (frac{sqrt{3}}{3})
45°	(frac{1}{sqrt{2}}) or (frac{sqrt{2}}{2})	(frac{1}{sqrt{2}}) or (frac{sqrt{2}}{2})	1
60°	(frac{sqrt{3}}{2})	(frac{1}{2})	(sqrt{3})

The Unit Circle and Trigonometric Ratios

The unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) of a coordinate plane. For any point ((x, y)) on the unit circle, the angle (theta) formed by the positive x-axis and the line segment from the origin to ((x, y)) has:

1. (cos theta = x) (the x-coordinate of the point)
2. (sin theta = y) (the y-coordinate of the point)

This relationship helps to visualize trigonometric ratios for angles beyond 90° and understand their periodic nature.

Graphs of Sine and Cosine Functions

The graphs of sine and cosine functions are periodic waves. They show how the values of (sin theta) and (cos theta) change as (theta) increases.

Graph of (y = sin x):

1. Starts at 0 at (x=0^circ).
2. Reaches maximum value of 1 at (x=90^circ).
3. Crosses 0 at (x=180^circ).
4. Reaches minimum value of -1 at (x=270^circ).
5. Crosses 0 at (x=360^circ).

Graph of (y = cos x):

1. Starts at 1 at (x=0^circ).
2. Crosses 0 at (x=90^circ).
3. Reaches minimum value of -1 at (x=180^circ).
4. Crosses 0 at (x=270^circ).
5. Reaches maximum value of 1 at (x=360^circ).

Both graphs have a period of 360° (or (2pi) radians) and an amplitude of 1.

Angle of Elevation and Angle of Depression

These terms describe angles formed by a horizontal line and a line of sight.

1. Angle of Elevation: This is the angle measured upwards from the horizontal line to the line of sight when an observer looks at an object above the horizontal.
2. Angle of Depression: This is the angle measured downwards from the horizontal line to the line of sight when an observer looks at an object below the horizontal.

It is important to note that the angle of elevation from point A to point B is equal to the angle of depression from point B to point A, assuming the horizontal lines are parallel.

Application of Special Angle Ratios to Simple Problems

Example 1

Question: A ladder 10 m long leans against a vertical wall. If the ladder makes an angle of 60° with the ground, how high up the wall does the ladder reach?

Solution:

Step 1: Draw a diagram and label the knowns and unknowns.

Let (h) be the height the ladder reaches up the wall. The ladder is the hypotenuse (10 m), and (h) is the opposite side to the 60° angle.

Step 2: Choose the appropriate trigonometric ratio.

Since we have the opposite side and the hypotenuse, we use the sine ratio: (sin theta = frac{text{Opposite}}{text{Hypotenuse}}).

Step 3: Substitute the values and solve.

(sin 60^circ = frac{h}{10})

From the special angles table, (sin 60^circ = frac{sqrt{3}}{2}).

(frac{sqrt{3}}{2} = frac{h}{10})

(h = 10 times frac{sqrt{3}}{2})

(h = 5sqrt{3})

Answer: The ladder reaches (5sqrt{3}) metres up the wall.

Example 2

Question: A man stands 15 metres away from the base of a tree. If the angle of elevation of the top of the tree from his position is 45°, what is the height of the tree?

Solution:

Step 1: Draw a diagram and label the knowns and unknowns.

Let (H) be the height of the tree. The distance from the man to the tree is the adjacent side (15 m), and (H) is the opposite side to the 45° angle.

Step 2: Choose the appropriate trigonometric ratio.

Since we have the opposite and adjacent sides, we use the tangent ratio: (tan theta = frac{text{Opposite}}{text{Adjacent}}).

Step 3: Substitute the values and solve.

(tan 45^circ = frac{H}{15})

From the special angles table, (tan 45^circ = 1).

(1 = frac{H}{15})

(H = 15 times 1)

(H = 15)

Answer: The height of the tree is 15 metres.

Application to Angle of Elevation and Depression Problems

Example 3 (Angle of Elevation)

Question: From a point on the ground 20 m away from the foot of a vertical pole, the angle of elevation of the top of the pole is 30°. Calculate the height of the pole.

Solution:

Step 1: Draw a right-angled triangle representing the situation.

Let the height of the pole be (h). The distance from the point to the pole is 20 m (adjacent). The angle of elevation is 30°.

Step 2: Identify the relevant trigonometric ratio.

We have the opposite side ((h)) and the adjacent side (20 m), so we use tangent.

(tan theta = frac{text{Opposite}}{text{Adjacent}})

Step 3: Substitute values and solve for (h).

(tan 30^circ = frac{h}{20})

We know (tan 30^circ = frac{1}{sqrt{3}}).

(frac{1}{sqrt{3}} = frac{h}{20})

(h = frac{20}{sqrt{3}})

To rationalize the denominator, multiply numerator and denominator by (sqrt{3}):

(h = frac{20sqrt{3}}{3})

Answer: The height of the pole is (frac{20sqrt{3}}{3}) metres.

Example 4 (Angle of Depression)

Question: A bird is perched on top of a 12 m tall tree. From the bird’s position, the angle of depression to a worm on the ground is 60°. How far is the worm from the base of the tree?

Solution:

Step 1: Draw a diagram.

Let the distance from the worm to the base of the tree be (x). The height of the tree is 12 m. The angle of depression from the bird to the worm is 60°. This means the angle of elevation from the worm to the bird is also 60° (alternate angles).

Step 2: Identify the relevant trigonometric ratio.

We have the opposite side (height of tree, 12 m) and the adjacent side (distance (x)), so we use tangent.

(tan theta = frac{text{Opposite}}{text{Adjacent}})

Step 3: Substitute values and solve for (x).

(tan 60^circ = frac{12}{x})

We know (tan 60^circ = sqrt{3}).

(sqrt{3} = frac{12}{x})

(x = frac{12}{sqrt{3}})

To rationalize the denominator:

(x = frac{12sqrt{3}}{3})

(x = 4sqrt{3})

Answer: The worm is (4sqrt{3}) metres from the base of the tree.

Teaching Methods/Instructional Techniques

Discussion, Demonstration, Guided Practice, Question and Answer, Explanation, Problem Solving, Pair Work, Group Work, Individual Practice

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Review/Questioning

Teacher’s Activity: The teacher greets the students and asks them to recall the basic trigonometric ratios (sine, cosine, tangent) and their values for special angles (30°, 45°, 60°). The teacher also asks about the components of a right-angled triangle.

Pupils’ Activity: Pupils respond by stating the ratios and special angle values, and identifying sides of a right-angled triangle.

Learning Point: Recall of trigonometric basics

Step 2: Review of Special Angles and Ratios

Time: 10 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher uses the cut-out shapes of right-angled triangles (30-60-90 and 45-45-90) to demonstrate how the special angle ratios are derived. The teacher then displays a chart showing the trigonometric ratios of special angles and guides students to brainstorm in groups to recall and verify these values without calculators.

Pupils’ Activity: Students observe the demonstration, participate in group brainstorming, and recall the special angle ratios.

Learning Point: Special angle ratio values

Step 3: The Unit Circle and Ratios

Time: 10 minutes

Teaching Skill: Explanation/Visualisation

Teacher’s Activity: The teacher uses the chart showing the unit circle to explain how sine and cosine ratios relate to the coordinates of a point on the unit circle. The teacher demonstrates how (x = cos theta) and (y = sin theta).

Pupils’ Activity: Pupils observe the chart, listen to the explanation, and ask questions for clarification.

Learning Point: Unit circle and ratios

Step 4: Graphs of Sine and Cosine

Time: 10 minutes

Teaching Skill: Demonstration/Guided Drawing

Teacher’s Activity: The teacher demonstrates how to draw the graphs of (y = sin x) and (y = cos x) for angles from 0° to 360° on the board, highlighting key points. Students are guided to sketch these graphs using rulers and pencils.

Pupils’ Activity: Pupils observe the demonstration and practice drawing the graphs in their notebooks.

Learning Point: Sine and cosine graphs

Step 5: Angle of Elevation and Depression

Time: 5 minutes

Teaching Skill: Explanation/Illustration

Teacher’s Activity: The teacher defines and illustrates the concepts of angle of elevation and angle of depression using clear diagrams on the board. The teacher emphasizes the horizontal line as the reference point and explains their relationship (alternate angles).

Pupils’ Activity: Pupils listen, observe the diagrams, and ask questions to understand the concepts.

Learning Point: Elevation and depression angles

Step 6: Problem Solving and Application

Time: 10 minutes

Teaching Skill: Problem Solving/Guided Practice

Teacher’s Activity: The teacher presents worked examples on the application of trigonometric ratios of special angles to simple problems, including those involving angles of elevation and depression. The teacher guides students through the steps, encouraging them to draw diagrams and select appropriate ratios. Students are encouraged to solve similar problems in pairs.

Pupils’ Activity: Pupils pay attention to the worked examples, participate in solving steps, and practice solving problems in pairs.

Learning Point: Solving application problems

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

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

1. What is the relationship between sine/cosine and the unit circle?
2. Sketch the graph of (y = sin x) for (0^circ le x le 360^circ).
3. Define angle of elevation and angle of depression.
4. A pole is 10 m high. From a point on the ground, the angle of elevation to the top of the pole is 60°. How far is the point from the base of the pole?

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Application of trigonometric ratios

Step 8: Note-Taking

Time: 10 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides pupils/students to copy the essential Board Summary notes, including the special angle table, unit circle concept, graph key points, and definitions of elevation/depression, into their notebooks.

Pupils’ Activity: Pupils/students copy the notes carefully into their notebooks.

Learning Point: Recording lesson content

Step 9: Conclusion

Time: 5 minutes

Teaching Skill: Consolidation

Teacher’s Activity: The teacher summarises the key learning points of the lesson, reinforcing the importance of understanding special angles and their application in real-world problems. The teacher encourages students to continue practicing problem-solving.

Pupils’ Activity: Pupils listen attentively and ask any final questions.

Learning Point: Lesson summary and reinforcement

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following problems in your notebook:

1. A kite is flying at a height of 50 m. The string attached to the kite makes an angle of 30° with the ground. Assuming the string is straight, find the length of the string.
2. From the top of a cliff 100 m high, the angle of depression of a boat at sea is 45°. How far is the boat from the foot of the cliff?
3. Draw the graph of (y = cos x) for (0^circ le x le 360^circ).

Lesson Keywords

• Trigonometry – The branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.
• Special Angles – Angles (like 30°, 45°, 60°) whose trigonometric ratios can be expressed exactly without a calculator.
• Unit Circle – A circle with a radius of 1 unit, centered at the origin, used to define trigonometric functions.
• Angle of Elevation – The angle measured upwards from the horizontal line to the line of sight.
• Angle of Depression – The angle measured downwards from the horizontal line to the line of sight.

Differentiation

For learners needing support: Provide additional practice with identifying opposite, adjacent, and hypotenuse sides. Offer a pre-filled table of special angle ratios for quick reference. Work through more step-by-step examples together.

For learners needing enrichment: Challenge them with problems involving two triangles or requiring multiple steps. Ask them to derive the special angle ratios using geometric constructions (equilateral triangle for 30/60, square for 45).

Suggested Lesson Videos

Search YouTube for “Trigonometric ratios of special angles SS1” or “Angle of elevation and depression problems SS1 Mathematics”.