Trigonometry and Euclidean Geometry ( Pythagorean Theorem)
Subject : Mathematics
Class : JSS 2
Term : Second Term
Week : Week 7
Topic : Trigonometry and Euclidean Geometry ( Pythagorean Theorem)
Trigonometry and Euclidean Geometry ( Pythagorean Theorem)
Previous Lesson :
Algebraic Fractions : Introduction and Expansion of Algebraic Expression
Objective: Students will learn about the Pythagorean Theorem and basic trigonometric functions, and how to apply these concepts to solve problems in Euclidean Geometry.
Materials:
- Whiteboard or chalkboard
- Markers or chalk
- Textbook with diagrams and practice problems
- Calculator (optional)
- Online Resources
Content ;
Trigonometry and Euclidean Geometry ( Pythagorean Theorem)
Euclidean Geometry is a type of geometry named after the ancient Greek mathematician Euclid. It’s the study of shapes, sizes, and positions of objects in space. One of the most famous theorems in Euclidean Geometry is the Pythagorean Theorem, which is used to find the length of the sides of a right-angled triangle.
A right-angled triangle is a triangle that has one angle of 90 degrees, also known as a right angle. The sides of a right-angled triangle are called the hypotenuse, the adjacent side, and the opposite side. The hypotenuse is the longest side of a right-angled triangle and is located opposite the right angle.
The Pythagorean Theorem tells us that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
In other words, c² = a² + b²
Here’s an example to help illustrate this:
Let’s say we have a right-angled triangle with sides of length a = 3 and b = 4. We want to find the length of the hypotenuse, which we’ll call c.
Using the Pythagorean Theorem, we know that:
c² = a² + b² c² = 3² + 4² c² = 9 + 16 c² = 25
To find c, we take the square root of both sides of the equation:
c = √25 c = 5
So the length of the hypotenuse is 5.
Now, let’s talk about Trigonometry:
Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It’s used in a variety of fields, such as engineering, physics, and navigation.
There are three main trigonometric functions: sine (sin), cosine (cos), and tangent (tan). These functions relate the ratios of the sides of a right-angled triangle to the angles in the triangle.
Let’s look at an example to help illustrate this:
Imagine we have a right-angled triangle with an angle of 30 degrees, and a side opposite that angle (labelled as “a” in the diagram) that has a length of 2. We want to find the length of the hypotenuse (labelled as “c” in the diagram).
To find c, we can use the trigonometric function of sine (sin):
sin(30 degrees) = opposite / hypotenuse sin(30 degrees) = a / c
We know that a = 2, so we can substitute that in:
sin(30 degrees) = 2 / c
We can rearrange the equation to solve for c:
c = 2 / sin(30 degrees)
Using a calculator, we can find that sin(30 degrees) = 0.5:
c = 2 / 0.5 c = 4
So the length of the hypotenuse is 4.
Trigonometry
There are three main trigonometric functions: sine (sin), cosine (cos), and tangent (tan). These functions relate the ratios of the sides of a right-angled triangle to the angles in the triangle.
Let’s look at an example to help illustrate this:
Imagine we have a right-angled triangle with an angle of 30 degrees, and a side opposite that angle (labelled as “a” in the diagram) that has a length of 2. We want to find the length of the hypotenuse (labelled as “c” in the diagram).
To find c, we can use the trigonometric function of sine (sin):
sin(30 degrees) = opposite / hypotenuse sin(30 degrees) = a / c
We know that a = 2, so we can substitute that in:
sin(30 degrees) = 2 / c
We can rearrange the equation to solve for c:
c = 2 / sin(30 degrees)
Using a calculator, we can find that sin(30 degrees) = 0.5:
c = 2 / 0.5 c = 4
So the length of the hypotenuse is 4.
Euclidean Geometry
Euclidean Geometry is a type of geometry named after the ancient Greek mathematician Euclid. It’s the study of shapes, sizes, and positions of objects in space. One of the most famous theorems in Euclidean Geometry is the Pythagorean Theorem, which is used to find the length of the sides of a right-angled triangle.
A right-angled triangle is a triangle that has one angle of 90 degrees, also known as a right angle. The sides of a right-angled triangle are called the hypotenuse, the adjacent side, and the opposite side. The hypotenuse is the longest side of a right-angled triangle and is located opposite the right angle.
The Pythagorean Theorem tells us that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
In other words, c² = a² + b²
Here’s an example to help illustrate this:
Let’s say we have a right-angled triangle with sides of length a = 3 and b = 4. We want to find the length of the hypotenuse, which we’ll call c.
Using the Pythagorean Theorem, we know that:
c² = a² + b² c² = 3² + 4² c² = 9 + 16 c² = 25
To find c, we take the square root of both sides of the equation:
c = √25 c = 5
So the length of the hypotenuse is 5.
The Pythagorean Theorem is used in many real-world applications, such as construction and architecture. It allows us to find the length of the diagonal of a rectangle, which is important in many building and design projects
Evaluation
- Which of the following is a right-angled triangle? a) Triangle with all sides equal b) Triangle with no equal sides c) Triangle with one angle of 90 degrees d) Triangle with all angles equal
Answer: c) Triangle with one angle of 90 degrees
- In a right-angled triangle, the longest side is called: a) The opposite side b) The adjacent side c) The hypotenuse d) The perpendicular side
Answer: c) The hypotenuse
- Which of the following trigonometric functions relates the opposite side to the hypotenuse in a right-angled triangle? a) Sine (sin) b) Cosine (cos) c) Tangent (tan) d) None of the above
Answer: a) Sine (sin)
- In a right-angled triangle, the ratio of the adjacent side to the hypotenuse is given by which of the following trigonometric functions? a) Sine (sin) b) Cosine (cos) c) Tangent (tan) d) None of the above
Answer: b) Cosine (cos)
- What is the Pythagorean Theorem used for? a) Finding the area of a circle b) Finding the length of the sides of a right-angled triangle c) Finding the volume of a sphere d) Finding the perimeter of a square
Answer: b) Finding the length of the sides of a right-angled triangle
- If a right-angled triangle has sides of length 3 and 4, what is the length of the hypotenuse? a) 5 b) 6 c) 7 d) 8
Answer: a) 5
- In a right-angled triangle, what is the relationship between the sine and cosine of an angle? a) They are equal b) They are opposite c) They are complementary d) None of the above
Answer: c) They are complementary
- What is the value of sin(30 degrees)? a) 0.5 b) 1 c) 0 d) -1
Answer: a) 0.5
- Which of the following is NOT a trigonometric function? a) Arcsine (arcsin) b) Arctangent (arctan) c) Cotangent (cot) d) Cosine-squared (cos²)
Answer: d) Cosine-squared (cos²)
- In a right-angled triangle, what is the relationship between the adjacent and opposite sides? a) They are equal b) They are complementary c) They are opposite d) None of the above
Answer: b) They are complementary
Lesson Presentation
Introduction: Begin by asking students if they’ve heard of the Pythagorean Theorem or trigonometry before. Briefly explain what each concept is and why they are important in mathematics and real-world applications. Ask students if they can think of any examples where these concepts might be used in real life.
Body:
- Pythagorean Theorem
- Define what a right-angled triangle is, and how to identify the hypotenuse, adjacent, and opposite sides.
- Explain what the Pythagorean Theorem is and how it can be used to find the length of the hypotenuse or other sides of a right-angled triangle.
- Work through some examples on the board, showing step-by-step how to apply the theorem to solve for a missing side.
- Give students some practice problems to work on individually or in small groups.
- Trigonometry
- Define the three main trigonometric functions: sine (sin), cosine (cos), and tangent (tan).
- Explain how each function relates the ratios of the sides of a right-angled triangle to the angles in the triangle.
- Work through some examples on the board, showing how to use trigonometric functions to solve for missing sides or angles in a right-angled triangle.
- Give students some practice problems to work on individually or in small groups.
- Applications
- Show students how the Pythagorean Theorem and trigonometry can be used in real-world applications, such as architecture, engineering, and navigation.
- Encourage students to think of other examples where these concepts might be used in real life, and discuss as a class.
Evaluation
- What is a right-angled triangle? Answer: A triangle that has one angle of 90 degrees.
- What is the hypotenuse of a right-angled triangle? Answer: The longest side of a right-angled triangle, located opposite the right angle.
- What is the Pythagorean Theorem used for? Answer: To find the length of the sides of a right-angled triangle.
- Can the Pythagorean Theorem be used in any triangle? Answer: No, it can only be used in right-angled triangles.
- What are the three main trigonometric functions? Answer: Sine (sin), cosine (cos), and tangent (tan).
- How does the sine function relate to a right-angled triangle? Answer: The sine function relates the ratio of the opposite side to the hypotenuse in a right-angled triangle.
- How does the cosine function relate to a right-angled triangle? Answer: The cosine function relates the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
- What is the relationship between the sine and cosine functions of an angle in a right-angled triangle? Answer: They are complementary.
- How can trigonometry be used in real life? Answer: Trigonometry is used in a variety of fields, such as engineering, physics, and navigation.
- What is an example of a real-life application of the Pythagorean Theorem? Answer: Finding the length of the diagonal of a rectangle, which is important in many building and design projects
Conclusion: Summarize the key concepts covered in the lesson, and ask students to share what they learned or found most interesting. Remind students of the importance of these concepts in mathematics and real-world applications, and encourage them to continue practicing and applying these skills.
Assessment: Assess students’ understanding through their participation in class discussions and group work, as well as their performance on individual practice problems. Consider giving a quiz or test at the end of the unit to further assess students’ understanding of the Pythagorean Theorem and basic trigonometric functions
Weekly Assessment /Test
- The Pythagorean Theorem is used to find the length of the sides of a __________ triangle.
Answer: right-angled
- The hypotenuse is the __________ side of a right-angled triangle.
Answer: longest
- The three main trigonometric functions are sine (sin), cosine (cos), and __________ (tan).
Answer: tangent
- The sine function relates the ratio of the __________ side to the hypotenuse in a right-angled triangle.
Answer: opposite
- The cosine function relates the ratio of the __________ side to the hypotenuse in a right-angled triangle.
Answer: adjacent
- The sine and cosine functions of an angle in a right-angled triangle are __________.
Answer: complementary
- Trigonometry is used in a variety of fields, such as engineering, physics, and __________.
Answer: navigation
- The Pythagorean Theorem allows us to find the length of the __________ of a rectangle.
Answer: diagonal
- In a right-angled triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the __________.
Answer: hypotenuse
- The Pythagorean Theorem can only be used in __________-angled triangles.
Answer: right
