Trig Identities, Part 2

Transition to slide 16. Give each student a copy of the Guided Notes handout.

Explain to the class that if they can recognize the style, or format, of a trigonometric equation, then determining the first step of solving is much easier. Ask students to reflect on the Appointment Clock activity and point out that they were able to solve all of these new trigonometric equations by relating them to a style of problem that they already knew how to solve.

Help students see that questions 1-2 from the Appointment Clock activity both use their knowledge of how to solve multi-step linear equations. Questions 3-4 both use their knowledge of solving a quadratic equation by taking a square root. Questions 5-6 both use their knowledge of factoring a quadratic expression. So, if they can recognize the format, “how” to solve the problem becomes much easier.

Direct students’ attention to example 1 on their Guided Notes handout. Complete example 1 as a class. Use the attached Guided Notes (Model Notes) document as needed for additional support and recommendations for the Guided Notes. Explain that factoring or using their knowledge of algebraic approaches—even the approaches they used during the Explore portion of this lesson—is often the first place to start.

Help students understand that example 1 has two terms that both have a common factor, so that is why factoring is the approach for solving this equation. Remind students that in the same way one should not divide both sides of an equation by x (potentially dividing by zero), one also should not divide both sides by a trigonometric expression.

Direct students’ attention to example 2. Ask if they think factoring or another algebraic approach would work to begin this problem. Explain that when using an algebraic method is not helpful for the first step, we ask ourselves: “Could I use a trigonometric identity?”

Challenge students to start example 2 on their own. After about 1 minute, giving everyone a chance to try the first step, ask for volunteers to share what they think the first step should be.

Once the class agrees on what the first step should be, have students complete example 2 on their own. While students work, monitor progress by circulating the room. Depending on time, write the steps on the board slowly so students can check their work as they go or have a volunteer go to the board to share their work.

Direct students’ attention to the back of their handout and emphasize the thought process of solving these problems by asking them the following guiding questions:

• What question should we ask ourselves first? Can we start algebraically?

• Is there an algebraic method that would be a helpful first step? No.

• What do we ask ourselves next? Can we use a trigonometric identity?

• Is there a trigonometric identity we can use? No.

Explain that what we do when this happens is that we make using a trigonometric identity an option by squaring both sides.

Remind students that just like solving radical equations, it is easiest to get one trigonometric equation isolated before squaring both sides. As a class, work through the first few steps together. After squaring both sides, point out that now the equation looks like example 2. Then direct students to complete example 3 on their own.

While students work, monitor progress by circulating the room. Depending on time, write the steps on the board slowly so students can check their work as they go or have a volunteer go to the board to share their work. Be sure to remind students that the process of squaring both sides of an equation increases the likelihood of extraneous solutions. So, it is important that they check their results when using this approach.

Once finished, have students add the handout to their math notebook if that is a classroom norm.