Created by Ethan Fahy

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1 Created by Ethan Fahy To proceed to the next slide click the button. Next

2 NCTM: Use trigonometric relationships to determine lengths and angle measures. NCTM: Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture. Michigan Department of Education: Differentiate and analyze classes of functions including linear, power, quadratic, exponential, and circular and trigonometric functions, and realize that many different situations can be modeled by a particular type of function. Michigan Department of Education: Use proportional reasoning and indirect measurements, including applications of trigonometric ratios, to measure inaccessible distances and to determine derived measures such as density. Find the sine, cosine, and tangent of an acute angle within a right triangle. Use trigonometric ratios to find side lengths in right triangles To apply trigonometric concepts in order to solve real-world problems. Next

3 How it Works: This activity will introduce the three geometric ratios (sine, cosine, tangent) and teach the relationships through a series of examples and activities. As you proceed through the PowerPoint you will have an opportunity to learn these concepts. There are tutorials, guided examples, and links to additional resources which should all be used before proceeding to the quiz. The quiz portion has three different difficulty levels (Beginner, Moderate, Expert). Once you can complete all three levels you will have a thorough understanding of these concepts. Audience: The intended audience is for any student enrolled in high school geometry. Key: During the slideshow the following links will always: Take you to the Home Page Take you to the Quiz Page Goal Of Learner: Recognize Geometric patterns and be able to apply them to right triangles and solve for missing sides. Next

4 The home page is the learning center for this activity. Each of the following links will take you to an informational page where you see definitions, examples, and applications to help you better understand the material. You will also find many links that will take you to additional resources to deepen your understanding. When you have completed each link click on the Quiz link. Vocabulary: Definitions of trigonometric ratios Click here if you need a calculator Special Right Triangles: Examples and Diagrams Finding Trig Ratios: Calculations, Examples, and Diagrams Link to: Multilingual Glossary Solving Right Triangles: Examples and Diagrams Calculating Trigonometric Ratios Quiz

5 Trigonometric Ratio is a ratio of two sides of a right triangle. There are three ratios that we will study. If you struggle remembering the ratios use this mnemonic device. Soh Cah Toa. Sin = Opposite Hypotenuse Cos = Adjacent Hypotenuse Tan = Opposite Adjacent Definition Sine: Ratio of the length of the leg opposite the angle to the length of the hypotenuse Notation Used: Sin Cosine: Ratio of the length of the leg adjacent the angle to the length of the hypotenuse Notation Used: Cos Tangent: Ratio of the length of the leg opposite the angle to the length f the leg adjacent to the angle. Notation Used: Tan Symbols Home Need more Explanation

6 By the AA Similarity Postulate, a right triangle with a given acute angle is similar to every other right triangle with that same acute angle measure. So ABC ~ DEF ~ XYZ, and. These are trigonometric ratios. A trigonometric ratio is a ratio of two sides of a right triangle. Home

7 Directions: Find Sin 30 o 1. Identify 30 o. 2. Label the sides of the triangle with relationship to 30 o. 3. Use the lengths to find the ratio opposite (across from angle) adjacent (side touching the angle) Link to: Guided Note Sheet In trigonometry, the letter of the vertex of the angle is often used to represent the measure of that angle. For example: Sine of A is written as SinA. Home

8 In trigonometry, we often use special right triangles to find exact values of trigonometry functions. The key to accomplishing this task is to 1. Recall each reference triangle and accurately construct it. 2. Identify the angle measure you will be using. 3. Recall the trigonometry ratio and fill in the values. Caution: Do not round answers if the directions ask for exact solutions Home

9 To find a length of a right triangle just follow these simple steps: 1. Identify the angle you are using 2. Label the sides of the triangle 3. Set up your ratio 4. Solve for the missing side Guided Example: Find the Length of BC (round your answer to the nearest tenth) opposite adjacent Caution: Do not round answers until the final step of your answer. Use the values of the trigonometric ratios provided by your calculator. BC = AC Tan15 o = BC = 10.2 ft Tan15 o = ft BC Home

10 To calculate a trigonometric ratio you just have to plug it into the calculator accurately. Caution: Be sure your calculator is in degree mode, not radian mode when computing trigonometric functions. On a graphing calculator it is under the mode button. Guided Ratio: Use your calculator to find the trigonometric ratio. Round to the nearest hundredth. cos76 o Cos(76) sin8 o Sin(8) cos(76 o )=0.24 sin(8 o )=7.12 Home

11 Beginner Click here if you need a calculator Expert Moderate Link to: Multilingual Glossary Home

12 Question 1 Question 3 Question 2 Move on to Moderate Quiz Home Quiz

13 What is the ratio for cos (cosine)? Opposite Hypotenuse Adjacent Hypotenuse Hypotenuse Opposite Quiz

14 What is the ratio for cos (cosine)? Remember what Soh - Cah Toa represents. Help

15 What is the ratio for cos (cosine)? Recall: Sin = Opposite Hypotenuse Cos = Tan = Opposite Adjacent Help

16 Trigonometric Ratio is a ratio of two sides of a right triangle. There are three ratios that we will study. If you struggle remembering the ratios use this mnemonic device. Soh Cah Toa. Sin = Opposite Hypotenuse Cos = Adjacent Hypotenuse Tan = Opposite Adjacent Definition Sine: Ratio of the length of the leg opposite the angle to the length of the hypotenuse Notation Used: Sin Cosine: Ratio of the length of the leg adjacent the angle to the length of the hypotenuse Notation Used: Cos Tangent: Ratio of the length of the leg opposite the angle to the length f the leg adjacent to the angle. Notation Used: Tan Symbols

17 You are ready to move on!! Quiz Next

18 Which trigonometric function would you use to find Opposite Adjacent Sin Cos Tan Quiz

19 Which trigonometric function would you use to find Opposite Adjacent Remember what Soh - Cah Toa represents. Help

22 Write the trigonometric ratio of sinj as a fraction Quiz

23 Write the trigonometric ratio of sinj as a fraction. adjacent Be sure to follow the steps and accurately label the sides of the triangle. opposite Help

24 Write the trigonometric ratio of sinj as a fraction. adjacent Recall: opposite Sin = Opposite Hypotenuse Help

27 Question 1 Question 3 Question 2 Move on to Expert Quiz Home Quiz

28 Calculate sin35 o. Round your answer to the nearest hundredth. Error Quiz

29 Calculate sin35 o. Round your answer to the nearest hundredth. Be sure you are typing in sin, cos, tan and NOT sin -1, cos -1, tan -1. The -1 represents the inverse function which is not what you have learned yet. Help

30 Calculate sin35 o. Round your answer to the nearest hundredth. Be sure you calculator is in degrees not radians. You need to check this every time you reset the memory on your calculator. Help

31 To calculate a trigonometric ratio you just have to plug it into the calculator accurately. Caution: Be sure your calculator is in degree mode, not radian mode when computing trigonometric functions. On a graphing calculator it is under the mode button. Guided Ratio: Use your calculator to find the trigonometric ratio. Round to the nearest hundredth. cos76 o Cos(76) sin8 o Sin(8) cos(76 o )=0.24 sin(8 o )=7.12

33 Given the following diagram: Find the cosb and write your answer as a decimal rounded to the nearest hundredth Quiz

34 opposite Given the following diagram: Find the cosb and write your answer as a decimal rounded to the nearest hundredth. adjacent Be sure to follow the steps and accurately label the sides of the triangle. Help

35 Given the following diagram: Find the cosb and write your answer as a decimal rounded to the nearest hundredth. The hypotenuse is always the longest side of a right triangle. So the denominator of a sine or cosine ratio is always greater than the numerator. Therefore the sine and cosine of an acute angle are always positive numbers less than 1. Since the tangent of an acute angle is the ratio of the lengths of the legs, it can have any value greater than 0. Help

36 Directions: Find Sin 30 o 1. Identify 30 o. 2. Label the sides of the triangle with relationship to 30 o. 3. Use the lengths to find the ratio In trigonometry, the letter of the vertex of the angle is often used to represent the measure of that angle. For example: Sine of A is written as SinA. Link to: Guided Note Sheet opposite (across from angle) adjacent (side touching the angle)

38 Use a special right triangle to write cos30 as a fraction. (Do not round your answer) Quiz

39 Use a special right triangle to write cos30 as a fraction. (Do not round your answer) Be sure that you construct the correct special right triangle Help

40 Use a special right triangle to write cos30 as a fraction. (Do not round your answer) Be sure you are using the appropriate relationship for cos (cosine): Cos = Adjacent Hypotenuse Help

41 In trigonometry, we often use special right triangles to find exact values of trigonometry functions. The key to accomplishing this task is to 1. Recall each reference triangle and accurately construct it. 2. Identify the angle measure you will be using. 3. Recall the trigonometry ratio and fill in the values. Caution: Do not round answers if the directions ask for exact solutions

43 Question 1 Question 3 Question 2 Home Quiz

44 Find the length of QR. Round to the nearest hundredth cm 2.16 cm 5.86 cm Quiz

45 Find the length of QR. Round to the nearest hundredth. Be sure you are using the appropriate steps: To find a length of a right triangle just follow these simple steps: 1. Identify the angle you are using 2. Label the sides of the triangle 3. Set up your ratio 4. Solve for the missing side Help

46 Find the length of QR. Round to the nearest hundredth. Be sure you are using the appropriate steps: To find a length of a right triangle just follow these simple steps: 1. Identify the angle you are using 2. Label the sides of the triangle 3. Set up your ratio 4. Solve for the missing side 63 o = X hypotenuse (always across from right angle) X 12.9 cm Help

47 To find a length of a right triangle just follow these simple steps: 1. Identify the angle you are using 2. Label the sides of the triangle 3. Set up your ratio 4. Solve for the missing side Guided Example: Find the Length of BC (round your answer to the nearest tenth) opposite adjacent Caution: Do not round answers until the final step of your answer. Use the values of the trigonometric ratios provided by your calculator. BC = AC Tan15 o = BC = 10.2 ft Tan15 o = ft BC

48 You have almost completed all the quizzes!! Quiz Next

49 Find the length ED. Round to the nearest hundredth cm cm cm Quiz

50 Find the length ED. Round to the nearest hundredth. Caution: Be sure your calculator is in degree mode, not radian mode when computing trigonometric functions. On a graphing calculator it is under the mode button. Make sure your measurements make sense in relationship to other sides of the triangle. Help

51 Find the length ED. Round to the nearest hundredth. Be sure you are using the appropriate steps: To find a length of a right triangle just follow these simple steps: 1. Identify the angle you are using 2. Label the sides of the triangle 3. Set up your ratio 4. Solve for the missing side tan 39 o = adjacent opposite X Help

52 To find a length of a right triangle just follow these simple steps: 1. Identify the angle you are using 2. Label the sides of the triangle 3. Set up your ratio 4. Solve for the missing side Guided Example: Find the Length of BC (round your answer to the nearest tenth) opposite adjacent Caution: Do not round answers until the final step of your answer. Use the values of the trigonometric ratios provided by your calculator. BC = AC Tan15 o = BC = 10.2 ft Tan15 o = ft BC

53 One more question to go!! Quiz Next

54 You are a contractor building a wheelchair ramp to replace stairs for one of the doorways into the school. The door is 1.8 feet above the ground. To meet the Americans with Disabilities Act the ramp from the ground to the door must be at a 2.86 o with the ground. To the nearest hundredth of a foot, what is the horizontal Distance covered by the ramp on the ground? feet 6.22 feet feet Quiz

55 You are a contractor building a wheelchair ramp to replace stairs for one of the doorways into the school. The door is 1.8 feet above the ground. To meet the Americans with Disabilities Act the ramp from the ground to the door must be at a 2.86 o with the ground. To the nearest hundredth of a foot, what is the horizontal Distance covered by the ramp on the ground? Caution: Be sure that you draw your diagram and label it accurately. This will help identify what you already know and what you are solving for. Be sure to fill in the numerical values. Help Username: mrfahy Password: geometry Page 528 Example 5 Ramp Height

56 You are a contractor building a wheelchair ramp to replace stairs for one of the doorways into the school. The door is 1.8 feet above the ground. To meet the Americans with Disabilities Act the ramp from the ground to the door must be at a 2.86 o with the ground. To the nearest hundredth of a foot, what is the horizontal Distance covered by the ramp on the ground? They are asking for the horizontal distance on the ground not the length of the actual ramp itself. Be sure to carefully read the directions. Help Username: mrfahy Password: geometry Page 528 Example 5

57 You Did It!! Click the link below if you would like to: Take Online Quiz with instant results More practice with application problems Home Quiz