Lesson 2: Introduction to Trigonometric ratios
Learning Goals
Lesson Aims
Introduction to Trigonometric ratios
Identifying trigonometric ratios in right-angled triangles
Learning Intentions
- Students define Trigonometric ratios, and identify the three ratios – sine, cosine, and tangent ratios
- Students use trigonometric notation $\sin \theta$, $\cos \theta$, $\tan \theta$ to identify the sine, cosine and tangent ratios in a right-angled triangle in terms of Opposite, Adjacent, and Hypotenuse.
Success Criteria
- Students are able to define trigonometric ratios as the relationship between sides and angles within a right-angled triangle.
- Students are able to describe sine, cosine, tangent ratios as the relationship between two sides with respect to a given angle.
- Students are able to recognise and use $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ to solve simple substitution questions.
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
- Define the sine, cosine, and tangent ratios for angles in right-angled triangles and use trigonometric notation $\sin \theta$, $\cos \theta$, $\tan \theta$
- Identify the sine, cosine, and tangent ratios in a right angled triangle
ICT and LIT
ICT
- Summary: Students explore and visualise sine, cosine, and tangent ratios in right-angled triangles using GeoGebra to understand relationships between sides and angles.
LIT
- Describe side relationships using mathematical terms (Opposite, Adjacent, Hypotenuse).
- Record observations from GeoGebra exploration using full sentences.
Prerequisite Knowledge
- Students should be able to identify the names of the sides relative to a given angle in a right angled triangle, using the exact terminologies: Opposite, Adjacent, Hypotenuse
- Students should be able to define Trigonometry and right-angled triangle
Resources
Whiteboard/Smartboard (markers), Student worksheet 2, Students’ devices, Presentation slides 2
Vocabulary List
- Trigonometric ratio – A ratio of two sides of a right-angled triangle with respect to a given angle
- Sine ($\boldsymbol{\sin \theta}$) – Opposite side divided by hypotenuse
- Cosine ($\boldsymbol{\cos \theta}$) – Adjacent side divided by hypotenuse
- Tangent ($\boldsymbol{\tan \theta}$) – Opposite side divided by adjacent side
- Ratio – A relationship between two numbers or quantities, showing how many times one value contains or is contained within the other
Lesson Structure
Introduction
20 minutes
- Greet and settle students down
- Give time for students to prepare for the lesson
- Students’ devices must be ready for use.
- Introduction to the concept: Trigonometric ratios
- Define ‘Trigonometric ratios’ as the relationships between sides and angles in triangles (slide 1)
- Teacher introduces three trigonometric ratios for learning: sine, cosine, tangent ratios (slide 2)
- Briefly explain the origins of these ratios, but mention that it is not assessable
- Teacher reminds students of the main focus
- Main focus of the topic is on right-angled triangles.
- Emphasise, therefore, that the ratios represent the relationship between two LENGTHS of sides in a right-angled triangle.
Body 1
Learning through Technology
25 minutes
- Students will learn visually through the use of technology, GeoGebra. (ICT)
- Give students 5 minutes to prepare devices and gain internet access if necessary.
- Teacher demonstrates how to gain access to the software, which program to search for, and how to use it
- Students follow the instructions carefully and gain access to the appropriate software and program.
https://www.geogebra.org/m/zfjbydbw
https://www.geogebra.org/m/ZJrYqevf
- Students follow the instructions carefully and gain access to the appropriate software and program.
- Allow students to explore and observe how the trigonometric ratios are visualised.
- Students are to pair up and discuss what they have observed, and what they have learnt. (Slide 3)
- Discussion questions
- What relationship do the trigonometric ratios represent in a right-angled triangle?
- If the size of the given angle stays constant, but one of the sides doubles, what happens to all other sides?
- GeoGebra Observation Writing (LIT): Students explore trigonometric ratios visually and write sentences describing patterns.
Body 2
Independent practice
10 minutes
- Students work independently on worksheet 2.
Math Explanation Writing (LIT): Students record calculations and describe which ratio was used and why.
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.