Today’s task involves a drawbridge and an investigation of what angle the bridge needs to be opened to in order for the bridge to be at a specified height above the water (as shown in the diagram). As a side note: this fictitious bridge is located in Kaliningrad, Russia. This city has a mathematical connection - can you or your students figure it out?
Give the task worksheet to students and see how far they can get. Even if students have experience with inverse trig functions, it will still be a challenging task. I urge you as a teacher to work through the task yourself before giving it to your students. There is a lot of math that is involved and you want to be prepared to deal with their struggles.
My students have used inverse trig before, but not as functions. They have just used inverses to find angle measures for right triangles in Geometry. Even if your students have never used any inverses before, this task is still a good preface to the discussion about inverse functions. If they cannot find the angle measure, identifying that they know the sine ratio and they need to know the angle measure is the most important part! You can go over the logistics of using your calculator to find the angle measure once they clear this conceptual hurdle.
Parts a-c of the worksheet should be pretty straightforward for students. When they get to part d, they will have to figure out the angle that the drawbridge should be open, based on the fact that the bridge will need to be open to a height of 17 feet. Students may use guess and check to find the angle. That is a great strategy and will be a good jumping off point to find a more efficient way. If students have never seen an inverse before, they probably will not have a more efficient strategy than guess and check, so as a class you can introduce the notation for inverse trig functions and instruct how to find the angle on the calculator.
After students have gotten through parts a-e, bring the class together to see their strategies for finding the angle measures that the bridge should open to. If guess and check was a common strategy, use this opportunity to help students realize that the input and outputs have switched – instead of knowing the angle measure and trying to find the sine ratio, we know the sine ratio and want to find the angle measure.
Ask students, "what happens when the input and output of a function switch places." Hopefully, they will remember that this procedure produces the inverse of a function. At this point, you can reacquaint your class with the notation for the inverse sine and go over what it does. Students will need the notation in order to write the function for parts f and g. Give them a couple minutes to work with their group to produce the function.
Sketching the graph of the function is going to be tricky. Before you let them go at it, I have a conversation about the domain and the range of the inverse function. We think about which x and y values would make sense in the context of this problem; that should get them on the right track to sketch. Students may plug in values or use their graphing calculator to sketch the graph. After giving them a few minutes, discuss as a class to make sure that everyone understands why the graph looks the way it does.
If students graphed on their graphing calculator or used a computer, they may wonder why the graph stops. Tell them that will discuss this more generally than just the context of the drawbridge. Ask students to think about the regular graph of y = sin x. Ask them what will happen if the inputs and outputs switch, like what happened during the task. Students may say that the graph will reflect over the line y = x. Ask them to sketch what that will look like. If they come up with the correct graph, they will see that it will not pass the vertical line test and is not a function.
Another way to think of this is if you have the equation y = sin-1(1). Ask them what that means so that they see that we are trying to find the angle y such that the sine of y is equal to 1. Ask a student for an answer and they will likely say 90°. Ask them if that is the only angle and another student will likely say that 450° will also work. Once they realize that there are an infinite number of angles that have a sine of 1, then they will see the need for the restriction for the inverse. We must restrict the range so that there is exactly one output for every input.
This is a really tough concept for students – the inverse trig functions become very abstract and students often lose sense of whether the output is the angle and the input is the ratio, or vice versa. Just like when we study logarithms and I continually remind them that a log is an exponent, for this unit I will continually talk about how an inverse trig output is an angle measure. In the video below is another strategy to make the distinction between the trig ratio and the angle measure.
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This is only the introduction as we will look at what happens with the other trig functions tomorrow. Students may have not mastered the concept today, but this task provides a good bridge (pun intended) from the conceptual to the procedural.