Today we will start off by building upon the Ferris wheel problems that students worked on yesterday for homework. In yesterday’s assignment, students were to create a graph that represented the distance from the ground as a function of time for a Ferris wheel that had a radius of 1 and went underground. Also, the length of the rotation was 360 seconds. Students essentially were using the unit circle and were calculating the sine values for where the rider would be after a specific amount of time. Since the length of one rotation was 360 seconds, calculating the rider’s height at 60 seconds, for example, corresponds to finding the sine of 60 degrees.
It is likely that many students in your class did not make the connection between the Ferris wheel problem and the unit circle. So we want them to make that connection with this discussion. Start with question A and ask students how they found the height of the rider at 90 seconds. Most will say that it is just the length of the radius. Now ask about question B and get a variety of responses. Students will likely have used Pythagorean Theorem or sine or cosine to find the vertical distance. Show both responses. Do the same for question C – survey your class to find the different methods for how students found the vertical height.
Now point out that the lines in the diagram could represent the x and y-axes. When they are finding the vertical distance of the rider, ask them how that relates to the ordered pair of the rider to see that we are really looking for the y-value to find the height. We want to generalize this relationship. Watch the video below to see how you can make the jump to the sine function.
After making the transition to talking about the sine and cosine, finish up the examples on the worksheet and talk about coterminal angles and how that makes sense in the context of the Ferris wheel problem.
Now that students have been reacquainted with the relationship between the sine and cosine and the unit circle, you want to solidify their knowledge. As I mentioned in yesterday’s lesson, my precalculus students have already studied radians and the unit circle, but I know they will need a quick refresher. If you are using this lesson to teach these concepts for the very first time, you may have to modify it to add a little more background and scaffolding.
Show students the PowerPoint and have them brainstorm about what they remember about trigonometry and the unit circle. We also want to review the six trigonometric functions and recall their definitions. I also want to review what a radian is. Radians are much more abstract than degrees and can be a difficult concept for students, so I definitely think it is worth it to spend the time talking about the concept even though they learned this last year. Students can often work with radians procedurally, but may have trouble explaining them conceptually.
Finally, we want to evaluate trigonometric expressions that involve angles and radians. Stress that students should be sketching out the angles in order to evaluate these expressions. Eventually they may get to the point that they can do them completely in their head. I always tell my students that knowing the trig values on the unit circle is similar to knowing your multiplication facts – you should have a conceptual understanding of how to evaluate, but there are tricks you can use to help figure out the values quickly. Here is a list of a few of these tricks:
1. Decide if it is a 30°-60°-90° triangle or a 45°-45°-90° triangle. Then associate the possible values with the triangle, sqrt(3)/2 and ½ or sqrt(2)/2, respectively. Choose the appropriate value and decide if it should be positive or negative.
2. Draw an accurate sketch of the unit circle and the given angle. Draw horizontal and vertical lines to the axes. If the line is really close to 1, then the value is sqrt(3)/2, if it is about halfway, the value is ½. If it is 45°, 135°, 225°, or 315°, then the values will always be plus or minus sqrt(2)/2.
After reviewing all of these concepts, an assignment will give students practice with these topics.