Lesson 10
Solving Problems with Trigonometry
10.1: Notice and Wonder: Practicing Perimeter (5 minutes)
Warmup
The purpose of this warmup is to elicit the idea that the more sides an inscribed polygon has the closer it comes to approximating a circle, which will be useful when students calculate perimeter and eventually \(\pi\) in later activities. While students may notice and wonder many things about these images, approximating a circle is the important discussion point.
This warmup prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).
Launch
Display the images for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a wholeclass discussion.
Student Facing
What do you notice? What do you wonder?
Student Response
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Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the word inscribed does not come up during the conversation, ask students if they remember the word and record the definition near a relevant item on the list of things students noticed. (We say a polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle. We say a circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.) Leave the responses visible throughout the lesson, as students will refer to them during the next activity and the lesson synthesis.
Demonstrate for students how the inscribed polygons change as the number of sides increases using the applet.
10.2: Growing Regular Polygons (20 minutes)
Activity
During this activity students are asked to compute the perimeter of inscribed regular polygons given only the radius. They will need to figure out that drawing in the altitude will form a right triangle. Drawing this auxiliary segment shows students recognize the important structure of right triangles to calculate missing information (MP7). In this activity students are building towards an equation to approximate the value of \(\pi\) which they will complete in the next lesson.
Launch
Provide the definitions of a regular polygon (a polygon where all of the sides are congruent and all the angles are congruent), pentagon (5sided polygon), and decagon (10sided polygon) as needed.
Supports accessibility for: Conceptual processing; Language
Student Facing

Here is a square inscribed in a circle with radius 1 meter. What is the perimeter of the square? Explain or show your reasoning.

What is the perimeter of a regular pentagon inscribed in a circle with radius 1 meter? Explain or show your reasoning.

What is the perimeter of a regular decagon inscribed in a circle with radius 1 meter? Explain or show your reasoning.

What is happening to the perimeter as the number of sides increases?
Student Response
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Student Facing
Are you ready for more?
Here is a diagram of a square inscribed in a circle and another circle inscribed in the same square.
 How much shorter is the perimeter of the small circle than the perimeter of the large circle?
 If the square was replaced with a regular polygon with more sides, would your previous answer be larger, smaller, or the same? Explain or show your reasoning.
Student Response
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Anticipated Misconceptions
If students are spending too long attempting to draw the pentagon or decagon, provide them with a circle that has 10 equally spaced points.
If students are stuck after sketching the diagram ask them what helpful auxiliary lines they could draw. (Connect the center to the other vertices to make triangles. The altitude would create right triangles so I can use trigonometry.)
Activity Synthesis
Invite students to share their strategies. Focus on students who repeated the same strategy for multiple questions.
Use the applet to demonstrate what happens with the perimeter of the inscribed polygon as the number of sides increases.
Ask students what they noticed. Connect the generalizations they make with the noticing and wondering they did in the warmup. If not mentioned by students, ask students what the circumference of the circle is. (\(2\pi\)) Build anticipation for calculating the value of \(\pi\) in the next lesson.
If students are not using workbooks tell them to be extra careful to put this work in a safe place because they will use it in the next lesson.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
10.3: Gentle Descent (10 minutes)
Activity
This activity requires more interpretation than a basic right triangle problem. First, students need to draw the diagram and figure out how to label it. Second, students need to recognize that the units do not match and figure out how to convert to a common unit. At this point the problem looks like a basic right triangle (with large numbers) and students will be ready to use their usual techniques of trigonometry and the Pythagorean Theorem.
Monitor for groups that convert to feet and groups that convert to miles.
Launch
If angle of descent isn’t familiar vocabulary, after 1 minute of quiet work time ask students which side will be longer, the horizontal or the vertical? (Horizontal because it is in miles.) Draw a diagram. Invite students to discuss how to label the diagram with important information including the angle of descent.
Supports accessibility for: Language; Socialemotional skills
Student Facing
An airplane travels 150 miles horizontally during a decrease of 35,000 feet vertically.
 What is the angle of descent?
 How long is the plane's path?
Student Response
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Anticipated Misconceptions
Prompt students to attend to precision with units.
Activity Synthesis
Invite the previously selected groups to share their process of converting to one of the common units. Acknowledge that both methods are valid. Ask which unit is clearer for answering the question of how far the plane traveled (miles).
Design Principle(s): Support sensemaking; Maximize metaawareness
Lesson Synthesis
Lesson Synthesis
Display the images and ask students “What do you notice? What do you wonder?” (All the images have circles. Some circles are inside the polygon and other circles are outside the polygon. How big are the polygons? Which is closest to the circumference of the circle?)
Use the applet to show a circle with a polygon both inside and outside the circle. Demonstrate increasing the number of sides of the polygons.
Add students’ observations and questions to the list from the warmup. Inform them they will be generalizing some of these ideas in the next lesson.
10.4: Cooldown  Again with Area (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
We know how to calculate the missing sides and angles of right triangles using trigonometric ratios and the Pythagorean Theorem. We can use the same strategies to solve some problems with other shapes. For example: Given a regular hexagon with side length 10 units, find its area.
Decompose the hexagon into 6 isosceles triangles. The angle at the center is \(360^\circ \div 6=60^\circ\). That means we created 6 equilateral triangles because the base angles of isosceles triangles are congruent.
To find the area of the hexagon, we can find the area of each triangle. Drawing in the altitude to find the height of the triangle creates a right triangle, so we can use trigonometry. In an isosceles (and an equilateral) triangle the altitude is also the angle bisector, so the angle is 30 degrees. That means \(\cos(30)=\frac{h}{10}\) so \(h\) is about 8.7 units. The area of one triangle is \(\frac12(10)(8.7)\), or 43.5 square units. So the area of the hexagon is 6 times that, or about 259.8 square units.