• Observations of the partner practice activity should focus on whether or not students assist each other in the identification of the appropriate measurements and their correspondence to the correct trigonometric ratios.
• In the Exit Ticket activity, use the correct answers to check the correspondences between measurements given and opposite sides, adjacent sides and hypotenuse, and opposite angles and adjacent angles.

Tell students, “Let’s say our community needs a new playground, and our class has been asked to design some of the equipment in the playground. We can’t just order equipment and show up at an area to begin building. There are regulations in place to keep the children who use the playground safe. For instance, playgrounds are not allowed to be built on cement surfaces because if a child falls, there is no cushion or absorption. The child may become more hurt than if the surface were softer. There are also regulations regarding the steepness of slides and teeter-totters. Slides are supposed to make a 30° angle or less with the ground and teeter-totters are supposed to make a 25° angle with the ground.” Show students the Picture of Slide handout (M-G-7-2_Picture of Slide.doc) with the images of the slide and teeter-totter. “Now let’s say these are the slide and teeter-totter we want to build for the community playground. How do we determine if the angle the slide makes with the ground is 30° or less, or if the angle the teeter-totter makes is 25°?” Give them a few minutes to think about it. Then show them the pictures of the slide and teeter-totter on the second page with the red lines drawn in. “What have I drawn in the picture?”

Part 1: Partner Practice

“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.” Hand out the Lesson 2 Graphic Organizer (M-G-7-2_Lesson 2 Graphic Organizer.doc and M-G-7-2_Lesson 2 Graphic Organizer KEY.doc) and have students take notes. Afterwards, hand out the Partner Practice Worksheet (M-G-7-2_Partner Practice Worksheet.doc and M-G-7-2_Partner Practice Worksheet KEY.doc). Students work in pairs on this worksheet. When they finish, have students meet with another pair to discuss the worksheet. Answer questions they may have.

Trigonometric Functions and Tables [IS.5 - All Students]

Note: Some mathematics textbooks contain reference tables for trigonometric functions. These tables are typically found in the back of the book in the same section as the index. Because scientific calculators are so efficient, widely available, and inexpensive, the use of trigonometry tables is not as common as it was a generation ago. Nevertheless, using trigonometry tables effectively is a useful skill, and understanding how the tables are structured can teach us something valuable about trigonometry.

Hand each student a copy of a trigonometric function table (M-G-7-2_Trigonometry Tables.docx).

“Let’s look at how this kind of table works. To find the value of a trigonometric ratio for a given angle, begin in the left margin of the table and locate the number of degrees of angle measure. Tables are generally presented with degrees in a vertical column in ascending order from 0 degrees to 90 degrees.”

“For example, to find the sine for an angle of 8 degrees, locate 8 in the degrees column. Move across that row to the right and find the column for Sine. In the Sine column, find the value 0.13917. This selection is equivalent to entering 8 in your calculator and pressing sin, result: 0.13917.”

“Now consider the complement of 8 degrees, 82 degrees. Find 82 in the degrees column, find the cosine column, and notice that the cosine of 82 degrees is the same as the sine of 8 degrees, 0.13917. Thinking about 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, since 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 note that the prefix co- suggests a particular 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, where both acute angles are 45 degrees, notice that the sine of 45° and the cosine of 45° are both equal (approximately 0.7071).”

“To find the angle measure for a given trigonometric ratio, start with the column for the selected ratio anywhere in the tables, and then move forward or backward, depending on the value of the ratio you are using. For example, to find the angle measure for the tangent = 0.80978, open the table to the tangent column and find any value. Move up or down the table in the tangent column to 0.80978, move to the left to find the corresponding number in the degrees column, 39 degrees.”

“If the trigonometric ratio is not exactly represented in the table, find the approximate equivalent by locating the ratio between two existing values. For example, to find the angle measure for the sine = 0.4, locate 0.39073 in the sine column for 23 degrees and locate 0.40674 in the sine column for 24 degrees. From these two values, you can infer that the angle for which the sine = 0.4 is between 23 and 24 degrees.” (In trigonometric tables that show intervals in minutes of one degree, you can locate the sine = 0.40008 at 23 degrees 35 minutes.)

Trigonometric Functions and Calculators

High school students are generally familiar with calculators and comfortable using them for simple operations. As with all student activities with calculators, emphasize practicing the procedures, writing down the steps, and checking the result to see it makes sense.

While calculators with trigonometric functions vary in operation, the typical procedure to find the value of one of the functions is to enter the number of degrees of the given angle, and then execute it by pressing the button for the required function. Ask students to get out their calculators.

“Let’s look at how to use a calculator with trigonometric functions. For example, to find the sine of 36 degrees, press 36, press sin, and the result appears in decimal form: 0.587785252…. That value can be used in a calculation to find the required side length by multiplying or dividing. To reverse the process and find the angle measure in degrees for a given trigonometric ratio, enter the trigonometric ratio for which you want to know the measure of the angle, press the inverse button (typically INV), and then press the button for the function.”

“For example, to find the measure of the angle for which the tangent is 0.525, enter 0.525, press INV, press tan, and the result shows 27.69947281.… The practical meaning of this result is that the ratio of the opposite side to the adjacent side of an angle of about 27.7 degrees in a right triangle is 0.525.”

Part 2

Hand out the Exit Ticket for this lesson (M-G-7-2_Lesson 2 Exit Ticket.doc and M-G-7-2_Lesson 2 Exit Ticket KEY.doc) to evaluate whether students understand the concept of trigonometric ratios.

Extension:

• Tell students that they are going to write the assessment 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 basic problems, or they can write real-world application problems. They are also to write the answer key for their six problems. Have students use their calculators and remind them to be selective in choosing which sides and angle measurements to use.

In Example 1, what is the length of the side, labeled x?

x and 9 represent the opposite side and hypotenuse, relative to the given angle. Therefore, we use the sin function. The necessary calculation steps are as follows:

In Example 2, what is the measure of x?

12 and 5 represent the adjacent and opposite sides, relative to the given angle. Therefore, we use the tan function and inverse tan function.

The necessary calculation steps are as follows:

                        Write the appropriate equation.

If the tangent of x = , what is x? Use the inverse tangent function.

       Evaluate and round to the nearest degree.