Using Half Angle Identities
Lesson 12 of 14
Objective: SWBAT find the half angle properties by rearranging the double angle properties then use the new properties to evaluate angles.
Today, I begin class by asking students to rearrange the double angle identity for sin u. This bellwork introduces a process that we will use to define the half-angle identities. After a few minutes I pick a student to show their work on the board. I pick different students each day in an effort to hold students accountable for completing work assigned in class.
I expect my students to struggle with rearranging the given equation. This is okay, because it will help to start a conversation. For students that are confused, I will eventually write the quadratic equation y = 1 - 2x^2 on the board for students to consider as an analogous problem.
When working with my students on this topic, I pay attention to what my students do when they take the inverse of a square. I ask "When you take the square root in order to solve for x, what are the possible outcomes?" This is usually enough of a reminder for my students. They know from past experience in Algebra 2 to take care when using a square root to rewrite an algebraic equation.
After our discussion of solving for sin u in the bell work, I will raise the idea with students that u is half of 2u. Then, I will encourage students to use this simple idea to find the half angle identity for sine by letting theta = 2u. With this instruction, I let my students get to work on the task of writing an expression for u (see resource). Once we have an expression for both u and 2u, we will substitute back into the rearranged formula from the bell work to produce the half angle identity for sine. Finally, students will record this identity on their reference sheets.
I now put up another form of the double angle identity for cosine. Students first have to solve for cos u. After solving for cos u the students will write a substitution for 2u and u. Students then find the half angle for sine and put this on their reference sheet.
For the half angle identities of sine and cosine, I plan to write the formula as sqrt(1/2(1+cos u)) because I find that my students have fewer issues evaluating this form of the identity. It also makes it easier for students to find the half angle identity for tangent. When it is time to find the half angle identity for tangent I plan to ask my students, "How might we find tangent using the identities we have already found?" The slide shows how a student worked the problem.
At the end of this lesson the students have all the identities I discuss in this class. Here is a typed version of the reference sheet. This was typed by a student and shared with the other students. Many students take their formula sheets and arrange the sheet so they can find the information quickly. Some of the advanced students take off some of the identities that they know and only put the identities they have not committed to memory.
I now want my students to practice using the half angle identities. I will begin by giving them a prompt. The students have done problems like this when they worked with the double angles, so I want to see how well they can apply their experiences on a related task. I plan to let the students work for about 3 minutes. Then, I will chose a student to share (see page 2 of using half angle).
One issue that I expect my students to have is how to determine if the sine and cosine will be positive or negative. I help students understand this using 2 different approaches:
- I write out the domain of theta and then divide all pieces of the domain by 2 which will tell them where theta/2 is located. So if theta is between [90,180]. Half of theta is between [45,90]. The students can then determine the sign of the half angle.
- I ask "Since our problem says theta is in quadrant II, what is the largest possible value theta could be?" Students say 179.99999 "Okay so it is less than 180, so what is the biggest angle that half theta could be?" A student will say it is less than 90. "What quadrant is an angle that is less than 90?"If I have an angle in quadrants III or IV I will also determine the smallest angle theta could be.
The next problem I have students work on is to find the exact value for cos 75. I discuss how theta/2=75 so theta=150. As shown on the slide the students replace the theta and theta/2 with the equivalent value. The students replace cos 150 with its value and then simplify the expression.
Students at first think we cannot have a square root inside a square root. I will discuss how sqrt(sqrt(3))=(3^1/2)^1/2 or 3^1/4. In this problem we leave the answer with 2 square roots since the equation has 2- sqrt(3).
To complete today's learning, I give students the following problems to complete:
p. 413, #45, 58, 68, 71 from Larson Precalculus with Limits
As class ends I ask the students to think of two different identities that can be used to find sin 15. Most student give one of two responses sin(45-30) or sin(30/2). We will look at these solutions in the bell work tomorrow since both methods are correct, but when evaluated the exact answers do not look the same.