Authentic Lessons for 21st Century Learning

Trig Identities, Part 2

Reciprocal, Quotient, and Pythagorean Identities

Michell Eike, Sherry Franklin | Published: September 18th, 2023 by K20 Center

  • Grade Level Grade Level 11th, 12th
  • Subject Subject Mathematics
  • Course Course Precalculus
  • Time Frame Time Frame 75-90 minutes
  • Duration More 2 class periods

Summary

Students will use the reciprocal, quotient, and Pythagorean trigonometric identities to solve equations. They will listen to a petroleum engineer describe his work and how trigonometric identities are used to solve problems when drilling. This lesson is the second lesson of four in a “Trig Identities” lesson series.

Essential Question(s)

How do we use trigonometric identities?

Snapshot

Engage

Students use the graphs of the sine and cosine functions to solve a trigonometric equation.

Explore

Students use their algebraic knowledge to solve trigonometric equations that are of the same format.

Explain

Students complete guided notes with the class and learn how to solve trigonometric equations using reciprocal, quotient, and Pythagorean identities.

Extend

Students listen to a petroleum engineer then apply what they have learned to solve a real-world problem.

Evaluate

Students demonstrate their understanding by independently solving a trigonometric equation.

Materials

  • Lesson Slides (attached)

  • Bell Ringer handout (attached; one half per student; printed front only)

  • Appointment Clocks handout (attached; one per student; printed front only)

  • Guided Notes handout (attached; one per student; printed front/back)

  • Guided Notes (Model Notes) document (attached; for teacher use)

  • Drilling for Filling handout (attached; one per pair; printed front only)

  • Drilling for Filling (Play-Doh) handout (optional; one per pair; printed front/back)

  • Oil Drilling handout (attached; one per pair; printed front/back)

  • Oil Drilling (Sample Responses) document (attached; for teacher use)

  • Exit Ticket handout (attached; one half per student; printed front only)

  • Pencil

  • Paper

  • Twinkies (one per student)

  • Plastic, bendy straws (2 per student)

  • Paper plates (1 per student)

  • Play-Doh (optional; one 3 oz. container per student)

  • Wax paper (optional)

Engage 1

5 Minute(s)

Use the Bell Ringer strategy to begin the lesson. As students enter the classroom, display slide 3 from the attached Lesson Slides and give each student a half-sheet of the attached Bell Ringer handout. Ask students to work independently to create the graphs and find the angle measure(s).

After giving students time to answer the question, transition to slide 4 so that students can check their work. Use this time to address any misconceptions. If time allows, facilitate a brief discussion on alternative, non-graphical approaches to solving the equation.

Go to slide 5 to share the lesson's essential question with students. Go to slide 6 to identify the lesson's learning objectives. Review each of these with students to the extent you feel necessary.

Explore

15 Minute(s)

Display slide 7 and give each student a copy of the attached Appointment Clocks handout. Use the slide to preview the activity. Explain that they will be using the Appointment Clocks strategy and will be working with 4 different partners to complete their handout. Use one of the handouts to help indicate to students to write the name of their partner for each appointment time in the corresponding box. Having students record their partner will help you know who worked with whom if you choose to collect this handout.

Transition to slide 8 and start the 3-minute timer on the slide. Direct students to find a partner and work together to solve questions 1-2 on their handout.

As students complete questions 1–2, transition to slide 9 so students can check their work. Give students time to correct their work and ask questions.

Display slide 10 and start the 5-minute timer on the slide. Direct students to find a new partner, and work together to solve questions 3–4 on their handout. Remind them to write their partner’s name on their handout for their 6 o’clock appointment.

As students complete questions 3–4, transition to slide 11 so students can check their work. Give students time to correct their work and ask questions.

Move to slide 12 and start the 5-minute timer on the slide. Direct students to find a new partner, and work together to solve questions 5–6 on their handout. Remind them to write their partner’s name on their handout for their 9 o’clock appointment.

As students complete questions 5–6, transition to slide 13 so students can check their work. Give students time to correct their work and ask questions.

Show slide 14 and direct students to find a new partner.

Transition to slide 15 and ask the class to discuss with their partner what was similar and what was different about the problems they solved. Start the 3-minute timer on the slide. As time allows, ask for a few volunteers to share with the whole class.

Explain

20 Minute(s)

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.

Engage 2

10 Minute(s)

Have students find a partner or assign students partners. Show slide 17 and read the following scenario: 

Petroleum engineers are professionals who specialize in the extraction of oil and gas from the Earth's crust. They use advanced drilling techniques to access underground reservoirs and retrieve these valuable resources. By analyzing geological data and employing innovative technology, petroleum engineers play a crucial role in locating and drilling wells to extract oil, ensuring a steady supply of energy for various industries and daily life.

Transition to slide 18 and ask students to discuss with their partner the question on the screen: Which method of drilling do you think will be more effective: vertical or horizontal? Why?

While students are discussing, pass out a copy of the attached Drilling for Filling handout to each pair.

Show slide 19 and direct their attention to their handout. Preview the activity with the class and explain that they will be using straws to drill for filling to see if vertical or horizontal drilling yields more filling. Direct students to gather the materials listed on their handout.

Display slide 20 and give the overview of the activity: each student should have two straws, one for vertical and one for horizontal drilling. When they extract their drilling devices (straws), they will place it on their data table, under the Drilling Records portion of their handout, and shade in the number of squares indicating the length of the filling extracted.

Transition through slides 21-24, reading the steps on each slide. Move through the slides at a pace that allows students time to complete the steps on the screen before moving to the next steps. 

Extend

20 Minute(s)

Show slide 25 and introduce the “Petroleum Engineering and Trigonometry” video, which features an interview with Derek Draper, a petroleum engineer, talking about his career and how trigonometry is used in his job.

After the video, transition to slide 26 and ask: What is an advantage of horizontal drilling? If needed, ask guiding questions or remind the class to think about the video and the Engage 2, Twinkie, activity to help them come to the conclusion that extracting more resources is a definite advantage of horizontal drilling over vertical drilling.

Give each student a copy of the attached Oil Drilling handout.

Transition through slides 27-29 and read the following scenario:

A well planner is working with a petroleum engineer to design a well. There will be three sections of drilling: the vertical portion, the curved portion, and the horizontal portion. The point where the drilling transitions from the vertical to curved portion is known as the kickoff point.

The drilling will need to transition from the vertical portion to the curved portion to bypass a salt dome section. Salt domes cause expensive challenges when drilling, so it is best if they are avoided.

The desired horizontal width, x, for the curved portion is 1,500 feet. The true vertical depth includes the vertical height of the curved portion, y, and vertical length of the vertical portion; this total needs to be 10,000 feet.

Display slide 30 and direct students’ attention to Part A of their handout. Direct students to discuss with their partner how they might start the problem.

Move through slides 31-32 and work through Part A as a class.

Show slide 33 and direct students’ attention to the back of their handout: Part B. Have students use their work from Part A to find θ.

To help with pacing, use slide 36 to direct their attention to Part C, finding the value of r. Then use slide 38 to direct their attention to Part D, calculating the measure depth. Lastly, use slide 40 to direct their attention to Part E, finding the buildup rate.

Evaluate

5 Minute(s)

Display slide 42 and use the Exit Ticket strategy to individually assess what students have learned from the lesson. Give each student a half-sheet of the attached Exit Ticket handout.

Collect student responses and use them to see which misconceptions persist before moving on to the next lesson: “Trig Identities, Part 3.”

Resources