Sum and Difference Day 2 of 2
Lesson 10 of 14
Objective: SWBAT use the sum and difference formulas to simplify expressions and prove identities.
Today's Bell Work asks my students to find two special angles whose sum or difference is 105 degrees. I pose the problem in degrees because I expect many of my students would convert to degrees, rather than work in radians (i.e., (7*pi)/24 is possible to work with, but not easy).
Teacher's Note: In my school it is important for students to work proficiently working with both degrees and radians. Our science courses work mainly in degrees, while our calculus course will work primarily in radians.
After letting students work for a couple of minutes, I bring the class together to discuss possible solutions. I expect some students to be confused by the prompt. I begin our discussion by asking, "What is meant by the term special angles?" Most will remember using the term special angles during our work with the unit circle. I am expecting some students found angles other than the multiples of 30 and 45. Once the meaning of the prompt is clarified, I ask students to take out their Unit Circle diagram. I want them to have this resource available to help us identify a number of different possible solutions.
The phrase "by adding or subtracting" makes this problem more interesting. While many students will immediately volunteer 45 degrees and 60 degrees, other students will have different responses. Page 2 of Bell Work Day 2 lists student responses.
As students offer examples I motivate discussion by asking questions like these:
- How did you choose the first angle?
- Is there more than one way to create a sum or different using a 135 degree angle?
- Can anyone find a result that will use an angle in quadrant IV?
I would like my students to recognize that they need one of the angles to be a multiple of 45 degrees, since that is our special angle ending in a 5. I work to get someone to notice this by looking at all the answers and asking what angle or triangle will be used to determine the trigonometric values.
Next, we build on our conversation from the Bell Work. I ask students to use the sum or difference formula to determine sin (105) by using one of the angle pairs from the Bell Work. As students work through this problem, I initially focus on whether or not students are making substitutions correctly.
- I encourage the students to begin by choosing an identity from their reference sheet.
- I suggest that they replace the A and B with their chose angle measures (e.g., 135 degrees and 30 degrees)
- I ask students to list the measures to be determined (e.g., sin 135, cos 30) and the exact values the measures.
On page 2 of sin 105 there are several examples of the values written as equations. I plan to use examples like these to ensure my students recognize that we are replacing sin (135), not just 135, with the trig measurement. I find that this mistake is common with my students. My students often write expressions like sin (1/2) or cos(sqrt2/2) when they make these substitutions because they are used to substituting for the argument to the function, rather than the function itself.
Once my students are on top of the necessary substitutions, most will complete the problem with success.
Teacher's Note: Our text, Larson's Precalculus with Limits, will factor out a sqrt 3 in most situations. I do not have the students do that since I am focusing on the process of evaluating. When students check answers as we are problem solving I will ask students to see if they can rewrite the answer in the book to be equivalent to their answer. This is a chance for students to work on number sense.
After exploring the use of the Sum and Difference formulas using degrees, we move on to radians. My students find it more difficult to work with radians.
- Radians are less familiar to them and they often lack a deep understanding of their meaning.
- Some of my students continue to prefer avoiding any work with fractions.
To help students get started with radians, I ask students to create a table with the special angles from the Unit Circle (see chart). I then announce that we will all find it easier to work with fractions for the next few problems. We'll discuss the benefit of using a common denominator of 12 and I give students a few minutes to write all of the values in their chart in 12ths. As they work, I expect one or more to ask, "How do you change pi or 2 * pi?" Most of the time, I can find a student who is willing to explain their process. I find this approach to be more helpful, when students help each other they own the knowledge.
Once we begin to work on problems, students may not, at first, understand how the chart makes things easier. But, as I give them angles in radians, and they use their chart to find the angles that add or subtract to the given angle, students become more and more confident in using the table. I make sure to give them both positive and negative angles, so that their confidence is challenged, then regained.
Here is an example of the work that the students will complete: sum radian example. After students find the angles that add or subtract to the given angle, students find the exact value of the trigonometric function for the angle in radians. Students write out all the substitutions and finally solve the problem as shown on page 2.
Correctly rewriting Sum and Difference Identities to evaluate them causes my students a lot of trouble. For example, my students often want to replace u and v with trigonometric values, instead of replacing the entire trig expression. I have found it helpful to have students explain what each part of the given problem means to them. I will ask numerous times as they work, "What do u and v represent in your work?"
I usually start this work by asking my students to find sin(u+v) with the given information. I continue to ask students to begin by writing out the identity. As the class works I ask the following questions:
- What information do we know from the prompt?
- What do we need to find?
- Is u=-4/5? What equals -4/5?
- If we know sin u how can we find cos u? (most of the time students want to draw triangles, even though they could use the Pythagorean identities to find cos u)
- Are angles u and v in the same triangle? How do you know?
Once students find the missing values of the trigonometric functions, they substitute and solve the equation. I will ask if the denominators of the 2 terms will always be the same? This is a way to check if the students are doing the problem correctly.
Another discussion we have is about labeling the diagrams the students make. I explain that when I forget to label I get confused about which diagram represents angle u and which represents angle v. At the end of this section of the lesson, I let the students work on a problem in their groups.
To close today's lesson, students are given time to work individually on problems. I assign my students some work from our textbook:
Page 402, # 7, 11, 13, 21, 43, 45, 47 from Larson's Precalculus with Limits
While students work I move around the room and check in with students who are struggling. I focus my attention on verifying students' procedures.
Very quickly, students will want to skip writing out the formula or the value of each trigonometric function in the equation. I strongly discourage this shortcut. Failing to write out the Identities at this stage causes students to make mistakes with substitution. Since this was an emphasis during the lesson, I move quickly when I see students taking shortcuts.