Lesson 28 Graph of the rotating arm

Episode 2

Reasoning Resources
Whole class preparation
Transferring length of the vertical projection to a single vertical axis with positive values 0-7.

Allowing pupils to extend the vertical axis into the negative domain.
Suggest that the hanging arm in the apparatus can be replaced with a vertical axis from the centre, starting from 0 in increments of 0.1 to 1. Demonstrate how the length of the hanging arm (which is the opposite side in a triangle) is really a vertical projection above the zero line. Show also that for all angles between 0 and 180° the vertical projection was between 0 and 1. Now start showing what happens when the turns continue.

Use the apparatus to show an angle greater than 180°. Discuss what the fact that the dangling arm is upside down means, leading the class to accept the need for negative values, Some may suggest extending the vertical axis downward, which you could allow them to do and scale giving them time to absorb the meaning.
Pair and small group work
Pupils continue with question 5 on Worksheet 1, finding the sines for ‘tens’ angles in the third and fourth quadrants (190, 200, etc.) and plotting them on the graph on Worksheet 2.
Relating the position of the hanging arm to negative values for the sines of angles greater than 180°.
Whole class sharing and discussion
Seeing from the graph that the sine ratio changes between +1 and -1.

Seeing the transformation as either a rotation by 180° or a double reflection.

Considering angles which are not usually met and relating the graph to the sine values of angles not on
the graph.
When most pupils have drawn the graph, bring the class back together to discuss what they have found. Develop this part of the lesson around the following questions:

• How would you describe this part (180°-360°) of the graph?
• What transformation is it from the first part of the graph?
• Where is the graph the steepest?
• What does this mean in terms of the sine ratio?
• What could an angle greater than 360° mean? A negative angle?,
• If tell you that the sine of an angle is 0.7, what can you tell me about the angle?
• How would you find the sine of 1000° from your graph?
• End of lesson reflection
Using the graph to find angles given the sine, and a sine value of a very large angle. In order to solve this pupils need to have grasped the cyclical nature of the graph.

Understanding that in the case of a unit circle the value of the sine could be read directly from the hanging arm. Linking sine and cosine with coordinates.

Beginning to understand the complementary nature of sine and cosine.
Suppose the rotating arm was length 1 and marked in tenths and the vertical arm was marked in tenths as well. What could you say about the length of the vertical arm for different angles? Imagine we have x- and y-axes through the centre of the circle which has a radius of 1 unit. At any point on the circle, which part of the coordinate would we read to give us the sine ratio?

You may want to extend this by looking at the other coordinate and relating this to the cosine of the angle. Plotting the cosine values and discussing the similarities and differences between the sine and cosine curves is a useful extension activity. The complementary relationship between sine and cosine can be explored using the angle whose sine is 0.5 (30°).Which angle has a cosine of 0.5? (60°). What is the relationship between these angles? Relate cosine to complementary angles. Several other avenues of extension are possible, all anchored on the concrete apparatus and accumulated constructions.