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Lesson overview
Lesson overview
In this lesson you will:
build on your understanding of similarity
explore the concepts of trigonometry
recall the characteristics of right-angled triangles
perform calculations with sine, cosine and tangent
Activity 1 - Building on similarity
Activity 1 - Building on similarity
Definition: Trigonometry is the mathematics of triangles. The name trigonometry comes from the Greek words trigonon, meaning triangle, and metron, meaning measure.
The concepts used in trigonometry build on what you have already learned about similarity.
Task 1 - Are they similar?
Task 1 - Are they similar?
Click on the image to open the Similarity and congruence page on the Math Interactives website.
Note: The page takes a little while to load so give it a minute.
Click on the tiny "bigger" button in the top right hand corner to make the interactive larger and easier to use.
Drag, rotate and resize the triangle on the left to compare it to the triangle on the right.
Determine whether the triangles are congruent, similar or neither.
Use the New button to get some more pairs to test.
What are similar right-angled triangles?
What are similar right-angled triangles?
These 2 right-angled triangles are similar. Think about why they are similar. Can you explain it?
What makes the triangles similar?
The two right-angled triangles above have an equal angle of 48°. We could show that the missing angle is equal by using the angle sum of a triangle.
This means that all of the angles in these triangles are equal. As a result, these right-angled triangles are similar.
Activity 2 - Exploring the characteristics of similar right-angled triangles
Activity 2 - Exploring the characteristics of similar right-angled triangles
In this activity, you will explore the characteristics of the side lengths of similar right-angled triangles.
Task 1 - Characteristics of the side lengths of similar right-angled triangles
Task 1 - Characteristics of the side lengths of similar right-angled triangles
Open the Desmos interactive, Exploring ratios of sides of right-angled triangles in a new tab by clicking on the image. (If the folder tab is open on the left hand side of the screen close it so you can see the graph in full screen.)
You can edit and change the triangles by dragging the black and purple dots on the vertices. This will create a new pair of similar right-angled triangles.
Copy the three tables from the Google Slide into your exercise book or folder.
Complete row 1 on Tables 1, 2 and 3, using information from the pair of triangles you can see in the Desmos activity.
Change the reference angle on one of the triangles in the Desmos interactive to create a new pair of triangles.
Record the new reference angle in the second row of your tables and then complete row 2 on Tables 1, 2 and 3.
Repeat steps 3 and 4 for 2 more pairs of triangles so that rows 3 and 4 are completed.
Task 2 - Summarising the characteristics of similar right-angled triangles
Task 2 - Summarising the characteristics of similar right-angled triangles
Activity 3 - Introducing sine, cosine and tangent
Activity 3 - Introducing sine, cosine and tangent
Right-angled triangles with the same reference angle are similar and the ratios of matching sides are equal.
These ratios can be determined using the functions Sine, Cosine and Tangent on a scientific calculator. These are known as sin, cos and tan for short.
Task 1 - Your calculator
Task 1 - Your calculator
Take a moment to locate these on your calculator.
Each of the functions gives the ratio for a different pair of side lengths.
Task 2 - Calculating sin, cos and tan
Task 2 - Calculating sin, cos and tan
Use your calculator and the tables you completed in Activity 2, Task 1, to complete the Google Doc activity.
Click on the button to open a new tab and view the Google Doc.
Click on the Use Template button to create a copy for you to edit.
Task 3 - Relating the ratios to the function
Task 3 - Relating the ratios to the function
Use the information from the tables in Activity 2, Task 1 to complete this interactive.
Handing in your work
Handing in your work
Don't forget to hand in the work you completed today!
Your teacher will have told you to do one of the following:
Upload any digital documents you created and any photos you took of your written work to your Learning Management system (MS Teams, Google Classroom for example).
Email any digital documents you created and any photos you took of your written work to your teacher.
Make sure you keep any hand written work you did in your exercise book or folder as your teacher may need to see these when you are back in class.
Student reflection
Student reflection
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