Summary
Students will explore inputs and outputs of trigonometric functions and use this relationship to identify the need for inverse trigonometric functions to find an unknown angle measure. Students will then watch a video of a civil engineer share how he uses right triangle trigonometry and apply their new knowledge to a real-world scenario. Students need to know how to find the sine, cosine, and tangent of an acute angle of a right triangle and know the basic trigonometric values of special angles (30°, 45°, and 60°) – or be able to sketch the triangle to find those trigonometric values – before beginning this lesson. This is the first lesson in the “Elevating Angles” lesson duo.
Essential Question(s)
How is trigonometry used to find the measures of unknown angles?
Snapshot
Engage
Students recall inputs and outputs of trigonometric functions.
Explore
Students compare the inputs and outputs of trigonometric functions with the inputs and outputs of inverse sine with a focus on special angles.
Explain
Students complete guided notes with the class and learn notation for inverse trigonometric functions and how to find the value of an unknown angle.
Extend
Students learn how a civil engineer uses trigonometry and then solve a real-world problem.
Evaluate
Students reflect on their learning using the What Did I Learn? strategy.
Materials
Lesson Slides (attached)
Inputs and Outputs handout (attached; one per group; printed front only)
Guided Notes handout (attached; one per student; printed front/back)
Go With the Flow handout (attached; one per pair; printed front only)
Scientific calculator (one per student)
Pencil
Paper
Engage
5 Minute(s)
Introduce the lesson using the attached Lesson Slides. Slide 3 displays the lesson’s essential question. Slide 4 identifies the lesson’s learning objective. Review each of these with your class to the extent you feel necessary.
Direct students into small groups of 2-3. Display slide 5 and introduce to the class the following questions: What are the inputs to a trigonometric function? What are the outputs? On the slide, students see an illustration of a function machine, where the input is on the top-left of the illustration and the output is in the bottom-right.
Transition to slide 6 and assign each group one trigonometric function to focus their discussion: sine, cosine, or tangent. This can be done a number of ways, for example, divide the room into thirds. Groups in the left third focus on the sine function, groups sitting in the center third focus on cosine, and groups in the right third focus on tangent.
Ask for a few volunteers to share what the input would be and what the output would be to a trigonometric function. Next, transition to slide 7. Use this slide to ensure that everyone understands that the input of the function is the angle and the output is the ratio of the lengths of two sides of a right triangle. Use student responses as a formative assessment to check if students are ready to proceed with the lesson or if they need to review sine, cosine, and tangent. They will be reviewing special angles during the next portion of the lesson, so a solid understanding is necessary.
Explore
15 Minute(s)
Show slide 8 and give each group a copy of the attached Inputs and Outputs handout. Remind students that special angles are commonly used in questions because they have simple inputs and outputs. Have groups complete their handout.
Once students complete their handouts, transition to slide 9. Remind students that they have been given an angle measure and asked to evaluate trigonometric functions. Ask them what they think they would do if they did not know the angle measure and wanted to find it. Have students write down any ideas on a piece of scratch paper at this time and do not yet share with the whole class. Give all students a chance to think on this.
After giving students time to think, click to display the next question on the slide. Ask the class a much simpler question: How do we find the value of x if x + 3 = 5? Here, allow volunteers to share out or call on students to respond. For students who respond with “Do the opposite of plus 3,” respond by asking them to use more academic language. Push students to use the word “inverse” operation.
Now, display slide 10 and introduce students to the I Notice, I Wonder strategy. Ask students to compare what they see on the slide with what is on their handout. Here, students see the inverse sine function evaluated for three ratios yielding three special angles. Do not yet use the vocabulary of “inverse sine.” In fact, do not even read the slide to the class.
Ask the class what they notice. Next, ask the class what they think sin–1( ) means by gesturing to the notation and referring to it as “that.” (What do you think “that” means?)
Explain
20 Minute(s)
Display slide 11. Give each student a copy of the attached Guided Notes handout and a scientific calculator.
Explain to the class how to read the notation. Note that negative one looks like an exponent, but is not an exponent. Explain to students that the inverse notation is not preferred by all, and that arcsine, arccosine, and arctangent are sometimes used instead. During Algebra 2, though, students will learn more about inverse functions and use that “negative 1 exponent-looking” notation, f –1(x), so this is the preferred notation to practice with for the time.
Take a moment to ask the class what the inputs and outputs of an inverse trigonometric function are and help tie it back to the function machine activity from the Engage portion of the lesson.
Complete example 1 as a class. Let students know that β (beta) is just another Greek letter, like θ, used to represent an angle. Students will also see α (alpha) later in the lesson.
Direct students’ attention to the back of their handout and introduce the vocabulary of angle of elevation, also known as angle of incline or angle of inclination. (Students will see angle of depression during the next lesson of this series.) Use slide 12 to either show students how to make notes on their handout or use it as a guide to demonstrate to students how to label their image on their handout.
Have students work independently or in pairs to find the angle of elevation in example 2. As students work, circulate the room. Ask for a volunteer to share what they found to be the angle and how they found it (the angle being approximately 34.992°). Consider having pairs of students share work by having one student verbally explain and the other write their algebraic work on the board.
Extend
30 Minute(s)
Display slide 13 and introduce the video on the slide: “K20 ICAP - Civil Engineering and Water Drainage.” The video introduces civil engineer Bobby Williams, who shares insights and advice about his profession. At the end of the video, Williams’ interview is linked to the real-world issue of preventing erosion. By using the right triangle trigonometry to determine value of the angle of elevation, it is possible to determine whether a ditch needs to be lined with concrete or grass.
Show slide 14 and give each student a copy of the attached Go With the Flow handout. Here, students are given four drainage ditch segments and are asked to find the angles of elevation. Next, they must determine if that segment needs a concrete liner or if it can remain grassed. If the angle of elevation is less than 0.3° or greater than 6°, then a concrete liner should be used.
Evaluate
5 Minute(s)
Move to slide 19. Use the What Did I Learn Today? strategy to have students reflect on their learning. On the back of their Go With the Flow handout, have students write 2-3 sentences and consider the following questions:
What did you learn that was new?
How was what you learned related to something you already knew?
What question(s) do you still have?
Collect students’ Go With the Flow handout to assess student learning and determine if students are ready for to the next lesson, “Elevating Angles, Part 2.”
Resources
K20 Center. (n.d.). What Did I Learn Today?. Strategies. https://learn.k20center.ou.edu/strategy/169
K20 Center. (n.d.). I Notice, I Wonder. Strategies. https://learn.k20center.ou.edu/strategy/180
K20 Center. (May 2023). K20 ICAP - Civil Engineering and Water Drainage. YouTube. https://youtu.be/JVUnXjCW2-o
