What do Triangles have to do with Circles?
Lesson 1 of 8
Objective: SWBAT apply knowledge of trigonometric ratios in right triangles to the unit circle. SWBAT use trigonometric ratios and the unit circle to solve problems.
Although my students have been keeping up their practice with trigonometry by solving right triangles in the Weekly Workout, I like to begin this lesson with a couple of warm-up problems. These problems are about finding lengths of lines in a circle (can you see where I'm going with this?) and will lead to some simple formulas. Some students may need a reminder that when a radius is drawn perpendicular to a chord it always bisects both the chord and the central angle the chord subtends (MP 8).
I will put the first problem on the board and ask everyone to begin solving it while I take attendance. After a couple of minutes, I'll ask, "How's it going?" (Hopefully, they'll say it was easy and they're all done.) I'll ask for a volunteer to talk me through the solution process while I act as a scribe at the board. At each step I'll check in with the rest of the class and answer any questions.
With the solution still on the board, we'll move to the next question: How do these lengths change if the angle were only half as big? In a minute or two the class should be ready to answer.
For the third and final problem I'll change the radius and ask the same question. By now, this problem has become pretty routine. Some students will already be thinking in terms of a formula. They might say, "It's easy, you just replace the 5 with a 10 here and here."
Having completed the three Opening Problems, it's time to discuss the implications. One way to begin is to ask, "Why does it seem simpler to find the distance from the center than the length of the chord?"
The answer is that in the case of the chord, we have to first find half of it, then double the result. There's an extra step in the process.
Ok, so it would be simpler if I just asked you to find the length of half the chord. I can do the doubling myself.
"Is there any other change we could make to simplify things?" Students aren't used to being able to change what's given, so they may need some prompting to see that the process would be simpler if they were given only half of the angle instead of the whole thing. (In my progression of diagrams on the whiteboard, you can see that I've redefined theta in the final diagram.)
Now, it's worth pointing out that what they've done is focus the entire problem on a single right triangle within the circle. Clever!
Now, I'll say, "There's just one simplification left to make with these formulas, isn't there? We already saw that circles of different sizes are similar - if you double the radius, the lengths of these lines double, too. So, if we know the lengths of these lines for one circle, we can easily adjust them for any other circle. What would be a convenient radius to use as our standard?" (For some thoughts on this reference to prior knowledge, see my reflection on this section.)
When the class recognizes that the ideal radius is the unit, I'll do a little dance for joy. We've arrived at the unit circle! Now it's time to bring the algebra back.
- Can anyone tell me the equation for the unit circle in the coordinate plane? (I sure hope so!)
- Why does the Pythagorean Theorem give us a circle? (The radius will be constant.)
- How long are these horizontal and vertical segments? (Cosine and sine of the angle.)
- What are the coordinates of the point on the circumference? What are they in terms of the angle?
As we discuss these questions, we'll gradually work our way up to this diagram of the unit circle. Once it's complete, I'll make the point that what we have here is an alternative definition for sine and cosine. In this case, sine and cosine aren't ratios, but lengths in a particular circle. I emphasize that these are equivalent definitions, and I like to give them explicitly side by side. And now we can see how intimately "triangle measurement" is related to circles (MP 7)!
[At this point, I love to tell my students the story of the strange names we use in trigonometry. You can read my outline of it here, but for the full story I highly recommend Trigonometric Delights, by Eli Maor.]
[This is also a great point to bring up the Pythagorean identities for the squares of sine and cosine. From a diagram like this one, they're self-evident!]
With this introduction to the unit circle, it's time for the students to gain some experience. These practice problems will help them to solidify their understanding of the relationship between the unit circle and the familiar right triangle trigonometry. They will be free to work in small groups, and I'll move around the room to make sure that everyone gets off to a good start.
These problems also require students to think about the sine and cosine of angles outside of the previous domain of 0 - 90 degrees. Bear in mind that while sin(120) makes perfect sense in the unit circle, it may seem like nonsense to students who are only familiar with right triangle trig. Be patient, help them to see the reference triangle in the second quadrant, and encourage them to make use of the new definitions (MP 6).
With about 5 minutes left in the period, I will call a brief halt to check our answers to the first 6 problems. I want to make sure that everyone has a chance to correct any mistakes and ask any necessary questions before class ends.
The final problems really push students to consider the possibility of extending the domain of the trig functions to include all real numbers. We'll discuss the implications of these problems in the next lesson. For now they are food for thought.
(As an aside, for your students with smart phones, you might recommend a unit circle app like this free one for your students. It's a handy thing to have at their fingertips. My students aren't allowed to have their phones with them in class, but it still might be helpful at home.)