Lesson 1: Introduction to Trigonometry
Learning Goals
Lesson Aims
Introduction to Trigonometry
Naming the sides of right-angled triangles
Learning Intentions
- Students define Trigonometry, and describe properties of a right-angled triangle
- Students identify the name of the sides in a right-angled triangle for a given angle
Success Criteria
- Students are able to define trigonometry as a study of measurements within triangles.
- Students are able to describe a right-angled triangle as a triangle with the right angle opposite to the longest side - the hypotenuse.
- Students are able to identify, for a given angle that is not the right angle, hypotenuse as the longest side of a right-angled triangle, opposite as the shorter side that does not make up the given angle, and adjacent as the shorter side that makes up the given angle.
Syllabus Links
MAO-WM-01
- develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
MA5-TRG-C-01
- Applies trigonometric ratios to solve right-angled triangle problems
Content descriptor
- Identify and label the hypotenuse, adjacent and opposite sides with respect to a given angle in a right-angled triangle in any orientation
ICT and LIT
ICT
None in this lesson
LIT
- Analyse the word “Trigonometry” and explain its meaning.
- Use mathematical vocabulary to describe relationships of sides relative to a given angle.
Prerequisite Knowledge
- Students should be able to apply Pythagoras’ theorem to solve problems in various contexts (MA4-PYT-C-01)
- Identify and describe the hypotenuse as the side opposite the right angle and the longest side in any right-angled triangle (Content descriptor)
Resources
Whiteboard/markers, Smartboard, Presentation slides 1, Student worksheet 1
Vocabulary List
- Trigonometry – The study of relationships between the angles and sides of triangles
- Right-angled triangle – A triangle with one angle exactly 90°
- Hypotenuse – The longest side of a right-angled triangle, opposite the right angle
- Opposite side – The side that is opposite a given angle in a triangle
- Adjacent side – The side that forms the given angle along with the hypotenuse in a right-angled triangle
- Angle ($\boldsymbol{\theta}$) – The figure formed by two rays with a common endpoint, often labelled as $\theta$ in trigonometry
- Sine ($\boldsymbol{\sin}$) – A trigonometric ratio defined as $\frac{\text{opposite}}{\text{hypotenuse}}$
- Cosine ($\boldsymbol{\cos}$) – A trigonometric ratio defined as $\frac{\text{adjacent}}{\text{hypotenuse}}$
- Tangent ($\boldsymbol{\tan}$) – A trigonometric ratio defined as $\frac{\text{opposite}}{\text{adjacent}}$
- Pythagoras’ theorem – In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides
Lesson Structure
Introduction
15 minutes
- Greet and settle students down
- Give time for students to prepare for the lesson
- This includes giving time to take out resources for the lesson
- Introduction to the topic: Trigonometry
- Vocabulary Analysis (LIT): Students break down “Trigonometry” into word roots (tri = three, gonia = angles, metry = measurement) and write one sentence explaining its meaning.
- Define ‘Trigonometry’ by drawing upon morphological analysis (Slides 1 and 2)
- Explicit mentioning of the focus triangle for the topic: Right-angled triangles (Slide 3)
- Teacher defines right-angled triangle
- A triangle with the right angle opposite to the longest side – the hypotenuse
Body 1
Concept teaching
10 minutes
- Teacher explains that the sides of a right-angled triangle have unique names relative to a given angle
- Slide 4: indicating given angle as $\theta$ with diagrams
- Ensure given angle is NOT the right angle.
- Teacher introduces the names of the sides with respect to given angle
- Slide 5: Draw upon students’ prior knowledge of Hypotenuse, and define Hypotenuse as the longest side of the right-angled triangle that is opposite the right angle
- Slide 6: Opposite side as the side that does NOT make up the given angle
- Slide 7: Adjacent side as the shorter side that makes up the given angle
NOTE: For ‘opposite’ side and ‘adjacent’ side, teachers can also define the side ‘opposite’ (on the other side) the given angle, and ‘adjacent’ (next to) the given angle.
Body 2
Guided practice
10 minutes
- Example Questions (Slides 7, 8, 9)
- Display various measurements of right-angled triangles with given angle $\theta$ (NOTE: All right-angled triangles must be in different orientation)
- Teacher guides students through the questions
- Clearly indicate the given angle
- Ensure names are introduced in the order of Hypotenuse, Opposite, and Adjacent to prevent confusion.
- Give time to students to think about the question before being guided through them.
- Make active use of call and response, prompt questions such as ‘why do you call this side _____?’ in order to maximise students’ understanding.
Body 3
Independent practice
15 minutes
- Students work independently on worksheet 1.
Math Explanation Writing (LIT): Students complete exercises on Worksheet 1, including written explanations for side names and relationships to given angle.
Students are expected to complete the exercise for homework if they have not finished in class
Teacher roams around the classroom to help individually students with questions.
- If many require explanation on a specific question, teacher brings attention from whole class to solve together the question on the board
Conclusion
5 minutes
Exit ticket
- Students are provided with a post-it note to draw any right-angled triangle and label the names of the sides for an angle of their choice.