<p>In this unit, students will study about trigonometric ratios. Students will:<br>- use trigonometric ratios to find a missing side of a right triangle.<br>- use trigonometric ratios to find a missing angle in a right triangle.&nbsp;<br>- use trigonometric ratios to solve real-world application problems.</p>

<p>- How would you explain the relationship between congruence and similarity in two and three dimensions?&nbsp;<br>- How are coordinates algebraically transformed to express, interpret, and validate geometric relationships?</p>

<p>- <strong>Angle:</strong> In geometry, the inclination to each other of two straight lines; the figure formed by two straight lines drawn from the same point, the vertex of the angle.&nbsp;<br>- <strong>Cosine:</strong> Abbreviated as cos; for a given angle is the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.<br>- <strong>Right Triangle:</strong> A triangle with one 90-degree angle.<br>- <strong>Sine:</strong> Abbreviated as sin; for a given angle is the ratio of the length of the side opposite to the length of the hypotenuse in a right triangle.<br>- <strong>Tangent:</strong> Abbreviated as tan; for a given angle is the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.<br>- <strong>Trigonometric Functions:</strong> Certain ratios of the sides of a right triangle containing an angle; are named sine, cosine, tangent, cotangent, secant, and cosecant.<br>- <strong>Trigonometric Ratio:</strong> In any right triangle, the quotient of the length of the opposite side divided by the length of the adjacent side is the tangent; the quotient of the length of the opposite side divided by the length of the hypotenuse is the sine; the quotient of the length of the adjacent side divided by the length of the hypotenuse is the cosine.<br>- <strong>Trigonometry:</strong> The name is derived from two Greek words meaning the study of triangles, and includes the study of the properties of triangles, including the solutions of sides and angles, and the functions that describe the various relationships between sides and angles.</p>

<p>- calculators (scientific or graphing)<br>- Picture of Slide handout (M-G-7-2_Picture of Slide)<br>- copies of the trigonometry tables (M-G-7-2_Trigonometry Tables)<br>- copies of the Lesson 2 Graphic Organizer (M-G-7-2_Lesson 2 Graphic Organizer and KEY)<br>- copies of Partner Practice Worksheet (M-G-7-2_Partner Practice Worksheet and KEY)<br>- copies of the Lesson 2 Exit Ticket (M-G-7-2_Lesson 2 Exit Ticket and KEY)</p>

<p>- Observations of the partner practice activity should focus on whether or not students help each other identify the relevant measures and their correspondence to the correct trigonometric ratio.&nbsp;<br>- In the Exit Ticket activity, utilize the correct answers to verify the correspondences between the given measures and opposite sides, adjacent sides and hypotenuse, and opposed angles and adjacent angles.</p>

<p>Modeling, Explicit Instruction<br>W: The lesson starts with a topic students can relate to: playgrounds. They discover that designing a playground requires careful consideration. There are rules and regulations for the equipment, and in this lesson, they help decide whether a slide and a teeter-totter should be put in a new playground in their neighborhood. They learn about trigonometric ratios and how they can be used to solve for missing sides and angles in right triangles. This lesson also shows some real-world applications of the trigonometric ratios. If there is time at the end of the lesson, students may write their own questions requiring the use of trig ratios.&nbsp;<br>H: The lesson begins with a hypothetical scenario involving the construction of a new playground in the town. Students have always utilized playgrounds, but they have probably never considered the amount of work and detail that goes into building them. They learn that slides and teeter-totters cannot be especially steep. This scenario teaches them to trigonometric ratios.&nbsp;<br>E: Many students struggle with trigonometric ratios. The graphic organizer shows them what the ratios are, why they are utilized, and how they are used. After following along and filling out the graphic organizer, students are given the opportunity to investigate trigonometric ratios with a partner. In the exit ticket, students decide whether the slide and teeter-totter in the pictures can be created in their new playground. Students have the ability to write their own problems.&nbsp;<br>R: The partner practice worksheet provides the most feedback on students' work. They first collaborate with a partner, then form a group of four to revise and refine their work. The exit ticket is beneficial for determining how well students understand a difficult topic and whether more time is required to study the information. Giving students the opportunity to write their own problems is another approach to determine how well they understand trig ratios.&nbsp;<br>E: Observe students as they work in pairs and groups. Discuss the problems with the class, and ask them to explain what they don't understand. The exit ticket and Extension Activity are ideal for evaluation. When reviewing the problems that students wrote, look for common mistakes such as the hypotenuse not being the longest side, using the incorrect trig ratio, or not utilizing the solving for x approach when x is in the denominator.&nbsp;<br>T: This lesson is suited for all students. The pace can be adjusted to accommodate different learner skill levels. Walking to the playground can help kinesthetic students learn about the angles and right triangles that exist all around us. Auditory learners are attracted in by the opening narrative and the partner practice worksheet when their partner reads the problems to them. Visual learners have a graphic organizer and can use drawings to solve problems.&nbsp;<br>O: The lesson begins with a relatable hypothetical situation. Students maintain the playground scenario in the back of their thoughts as they study trigonometric ratios. They practice with trig ratios, and at the end of the class, they use trig ratios to see if a slide and a teeter-totter can be built in the hypothetical playground.</p>

<p>Tell students,<strong> "Suppose our community needs a new playground, and our class has been tasked with designing some of the equipment in the playground. We can't simply order equipment and show up to a site to start building. There are regulations in place to keep children who use the playground safe. For example, playgrounds are not permitted to be built on cement surfaces since there is no cushioning or absorption in case a child falls. The child may sustain greater damage than if the surface were softer. There are additional regulations for the steepness of slides and teeter-totters. Slides are required to make a 30° or less angle with the ground, whereas teeter-totters are required to make a 25° angle with the ground."</strong> Show students the Picture of Slide handout (M-G-7-2_Picture of Slide), which includes photographs of the slide and teeter-totter. <strong>"Suppose these are the slides and teeter-totters we want to build for the community playground. How can we know if the slide's angle with the ground is 30° or less, or if the teeter-totter's angle is 25°?" </strong>Allow them a few minutes to think about it. Then show them the photos of the slide and teeter-totter on the second page, with the red lines drawn. <strong>"What have I drawn in the picture?"</strong></p><p><strong>Part 1: Partner Practice.</strong></p><p><strong>"In this lesson, we are going to learn how to solve for that angle as well as if we needed to solve for a missing side."</strong> Give out the Lesson 2 Graphic Organizer (M-G-7-2_Lesson 2 Graphic Organizer and KEY) and ask students to take notes. Next, distribute the Partner Practice Worksheet (M-G-7-2_Partner Practice Worksheet and KEY). Students work on this task in pairs. When they are finished, have the students meet with another pair to discuss the worksheet. Answer any questions they may have.</p><p><strong>Trigonometric Functions and Tables</strong></p><p><strong>Note: </strong>Some mathematics textbooks include reference tables for trigonometric functions. These tables are usually placed at the back of the book, in the same section as the index. Because scientific calculators are so efficient, widely available, and inexpensive, trigonometry tables are less commonly used than they were a generation ago. Nevertheless, using trigonometry tables effectively is an important skill, and understanding how the tables are organized can teach us a lot about trigonometry.</p><p>Give each student a copy of the trigonometric function table (M-G-7-2_Trigonometry Tables).</p><p><strong>"Let's see how this kind of table works. To find the value of a trigonometric ratio for a particular angle, start at the left border of the table and look for the number of degrees of angle measurement. Tables typically display degrees in a vertical column in ascending order from 0 degrees to 90 degrees."</strong></p><p><strong>"For example, to determine the sine of an angle of 8 degrees, find 8 in the degrees column. Move over that row to the right to find the Sine column. In the Sine column, find the value 0.13917. This selection is equivalent to entering 8 into your calculator and pressing sin; the result is 0.13917."</strong></p><p><strong>"Consider the complement of 8 degrees: 82 degrees. Locate 82 in the degrees column, then the cosine column, and note that the cosine of 82 degrees is the same as the sine of 8 degrees, 0.13917. When considering a right triangle with an 8-degree angle and an 82-degree angle, it makes sense that the cosine of one is the sine of the other, as the sine is the ratio of the length of the opposite side to the hypotenuse and the cosine is the length of the adjacent side to the hypotenuse. Also, the prefix co- indicates a certain relationship between the sine of an angle and the cosine of the same angle. The sine of an angle is equal to the cosine of its complement and similarly, the cosine of an angle is equal to the sine of its complement. In an isosceles right triangle with both acute angles at 45 degrees, notice that the sine and cosine of 45 degrees are both equal (about 0.7071)."</strong></p><p><strong>"To find the angle measure for a specific trigonometric ratio, begin with the column for the selected ratio anywhere in the tables and then move forward or backward, depending on the ratio's value. For example, to determine the angle measure for the tangent = 0.80978, open the table to the tangent column and select any value. Move up or down the table in the tangent column to 0.80978, and then to the left to find the corresponding value in the degrees column, 39 degrees."</strong></p><p><strong>"If the trigonometric ratio is not precisely represented in the table, obtain an approximate equivalent by calculating the ratio of two existing values. For example, to calculate the angle measure for sine = 0.4, find 0.39073 in the sine column for 23 degrees and 0.40674 in the sine column for 24 degrees. These two values indicate that the angle for which the sine equals 0.4 is between 23 and 24 degrees." </strong>(In trigonometric tables that display intervals in minutes of one degree, you can find sine = 0.40008 at 23 degrees 35 minutes.)</p><p><strong>Trigonometric Functions and Calculators</strong></p><p>High school students are generally familiar with calculators and can use them for simple operations. As with all student activities with calculators, encourage practicing the techniques, writing down the steps, and double-checking the results to ensure they make sense.</p><p>While the operation of trigonometric calculators varies, the common technique for determining the value of one of the functions is to enter the number of degrees of the given angle and then execute it by hitting the appropriate function button. Ask students to take out their calculators.</p><p><strong>"Let's look at how to utilize a calculator for trigonometric functions. To calculate the sine of 36 degrees, press 36, then press </strong><i><strong>sin</strong></i><strong>, and the answer is displayed in decimal form: 0.587785252.... That value can be used to calculate the required side length by multiplying or dividing. To reverse the process and determine the angle measure in degrees for a particular trigonometric ratio, enter the trigonometric ratio, hit the inverse button (usually INV), and then the function button."</strong></p><p><strong>"For example, to calculate the angle with tangent 0.525, input 0.525, press INV, and then press </strong><i><strong>tan</strong></i><strong> to get 27.69947281..... This practical meaning of this result is that the ratio of the opposite side to the adjacent side of a right triangle with an angle of approximately 27.7 degrees is 0.525."</strong></p><p><strong>Part 2</strong></p><p>Distribute the Exit Ticket for this lesson (M-G-7-2_Lesson 2 Exit Ticket and KEY) to assess students' understanding of trigonometric ratios.</p><p><strong>Extension:</strong></p><p>Tell students that they will be writing an evaluation on trigonometric ratios. They should write six problems, three in which there is a missing side, and three in which there is a missing angle. They can write either simple or real-world application challenges. They must also write the answer key for their six problems. Allow students to use their calculators and encourage them to be selective about the sides and angle measurements to use.</p><figure class="image"><img style="aspect-ratio:341/132;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_175.png" width="341" height="132"></figure><figure class="image"><img style="aspect-ratio:229/235;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_176.png" width="229" height="235"></figure><p>In Example 1, what is the length of the side labeled <i>x</i>?</p><p><i>x</i> and 9 denote the opposing side and hypotenuse of the given angle. Therefore, we use the sin function. The computation steps are as follows:</p><figure class="image"><img style="aspect-ratio:423/132;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_76.png" width="423" height="132"></figure><p>In Example 2, what is the measurement of <i>x</i>?</p><p>12 and 5 indicate the adjacent and opposite sides of the specified angle. As a result, we use both the tan and inverse tan functions.</p><p>The computation steps are as follows:</p><p>tan x = \(5 \over 12\) &nbsp; &nbsp; &nbsp; &nbsp; Write the relevant equation.</p><p><br>If the tangent of <i>x</i> = \(5 \over 12\), what is <i>x</i>? Apply the inverse tangent function.</p><p>\(5 \over 12\) ≈ 0.417 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Evaluate and round to the nearest degree.<br>tan (22.6°) ≈ 0.417<br>x ≈ 23°</p>