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Trigonometry for Measuring Indirect Distances in Real World

Learn how to use trigonometry to calculate the three basic trigonometric functions - sine, cosine, and tangent - for measuring indirect distances. Includes key concepts, common errors, and guided practice examples.

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Trigonometry for Measuring Indirect Distances in Real World

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  1. Introduction In the real world, if you needed to verify the size of a television, you could get some measuring tools and hold them up to the television to determine that the TV was advertised at the correct size. Imagine, however, that you are a fact checker for The Guinness Book of World Records. It is your job to verify that the tallest building in the world is in fact BurjKhalifa, located in Dubai. Could you use measuring tools to determine the size of a building so large? It would be extremely difficult and impractical to do so. 2.2.1: Calculating Sine, Cosine, and Tangent

  2. Introduction, continued You can use measuring tools for direct measurements of distance, but you can use trigonometry to find indirect measurements. First, though, you must be able to calculate the three basic trigonometric functions that will be applied to finding those larger distances. Specifically, we are going study and practice calculating the sine, cosine, and tangent functions of right triangles as preparation for measuring indirect distances. 2.2.1: Calculating Sine, Cosine, and Tangent

  3. Key Concepts The three basic trigonometric ratios are ratios of the side lengths of a right triangle with respect to one of its acute angles. As you learned previously: Given the angle Given the angle Given the angle 2.2.1: Calculating Sine, Cosine, and Tangent

  4. Key Concepts, continued Notice that the trigonometric ratios contain three unknowns: the angle measure and two side lengths. Given an acute angle of a right triangle and the measure of one of its side lengths, use sine, cosine, or tangent to find another side. Use the inverses of these trigonometric functions (sin–1, cos–1, and tan–1) to find the acute angle measures given two sides of the right triangle. 2.2.1: Calculating Sine, Cosine, and Tangent

  5. Common Errors/Misconceptions not using the given angle as the guide in determining which side is opposite or adjacent forgetting to ensure the calculator is in degree mode before completing the operations using the trigonometric function instead of the inverse trigonometric function when calculating the acute angle measure 2.2.1: Calculating Sine, Cosine, and Tangent

  6. Guided Practice Example 2 A trucker drives 1,027 feet up a hill that has a constant slope. When the trucker reaches the top of the hill, he has traveled a horizontal distance of 990 feet. At what angle did the trucker drive to reach the top? Round your answer to the nearest degree. 2.2.1: Calculating Sine, Cosine, and Tangent

  7. Guided Practice 2.2.1: Calculating Sine, Cosine, and Tangent

  8. Guided Practice: Example 2, continued Determine which trigonometric function to use by identifying the given information. Given an angle of w°, the horizontal distance, 990 feet, is adjacent to the angle. The distance traveled by the trucker is the hypotenuse since it is opposite the right angle of the triangle. 2.2.1: Calculating Sine, Cosine, and Tangent

  9. Guided Practice: Example 2, continued Cosine is the trigonometric function that uses adjacent and hypotenuse, so we will use it to calculate the angle the truck drove to reach the bottom of the road. 2.2.1: Calculating Sine, Cosine, and Tangent

  10. Guided Practice: Example 2, continued Set up an equation using the cosine function and the given measurements. Therefore, Solve for w. 2.2.1: Calculating Sine, Cosine, and Tangent

  11. Guided Practice: Example 2, continued Solve for w by using the inverse cosine since we are finding an angle instead of a side length. 2.2.1: Calculating Sine, Cosine, and Tangent

  12. Guided Practice: Example 2, continued Use a calculator to calculate the value of w. On a TI-83/84: First, make sure your calculator is in DEGREE mode: Step 1: Press [MODE]. Step 2: Arrow down twice to RADIAN. Press [ENTER]. Step 3: Arrow right to DEGREE. Step 4: Press [ENTER]. The word “DEGREE” should be highlighted inside a black rectangle. 2.2.1: Calculating Sine, Cosine, and Tangent

  13. Guided Practice: Example 2, continued Step 5: Press [2ND]. Step 6: Press [MODE] to QUIT. Note: You will not have to change to DEGREE mode again unless you have changed your calculator to RADIAN mode. Next, perform the calculation. Step 1: Press [2ND][COS][990][÷][1027][)]. Step 2: Press [ENTER]. w = 15.426, or 15°. 2.2.1: Calculating Sine, Cosine, and Tangent

  14. Guided Practice: Example 2, continued On a TI-Nspire: First, make sure the calculator is in degree mode: Step 1: Choose 5: Settings & Status, then 2: Settings, and 2: Graphs and Geometry. Step 2: Move to the Geometry Angle field and choose “Degree”. Step 3: Press [tab] to “ok” and press [enter]. 2.2.1: Calculating Sine, Cosine, and Tangent

  15. Guided Practice: Example 2, continued Next, perform the calculation. Step 1: In the calculate window from the home screen, press [cos–1][990][÷][1027]. Step 2: Press [enter]. w = 15.426, or 15° The trucker drove at an angle of 15° to the top of the hill. ✔ 2.2.1: Calculating Sine, Cosine, and Tangent

  16. Guided Practice: Example 2, continued 2.2.1: Calculating Sine, Cosine, and Tangent

  17. Guided Practice Example 4 Solve the right triangle. Round sides to the nearest thousandth. 2.2.1: Calculating Sine, Cosine, and Tangent

  18. Guided Practice: Example 4, continued Find the measures of and . Solving the right triangle means to find all the missing angle measures and all the missing side lengths. The given angle is 64.5° and 17 is the length of the adjacent side. With this information, we could either use cosine or tangent since both functions’ ratios include the adjacent side of a right triangle. Start by using the tangent function to find . 2.2.1: Calculating Sine, Cosine, and Tangent

  19. Guided Practice: Example 4, continued Recall that 2.2.1: Calculating Sine, Cosine, and Tangent

  20. Guided Practice: Example 4, continued On a TI-83/84: Step 1: Press [17][TAN][64.5][)]. Step 2: Press [ENTER]. x = 35.641 On a TI-Nspire: Step 1: In the calculate window from the home screen, press [17][tan][64.5]. Step 2: Press [enter]. x = 35.641 The measure of AC = 35.641. 2.2.1: Calculating Sine, Cosine, and Tangent

  21. Guided Practice: Example 4, continued To find the measure of , either acute angle may be used as a reference. Since two side lengths are known, the Pythagorean Theorem may be used as well. Note: It is more precise to use the given values instead of approximated values. 2.2.1: Calculating Sine, Cosine, and Tangent

  22. Guided Practice: Example 4, continued Use the cosine function based on the given information. Recall that 2.2.1: Calculating Sine, Cosine, and Tangent

  23. Guided Practice: Example 4, continued 2.2.1: Calculating Sine, Cosine, and Tangent

  24. Guided Practice: Example 4, continued On a TI-83/84: Check to make sure your calculator is in DEGREE mode first. Refer to the directions in the previous example. Step 1: Press [17][÷][COS][64.5][)]. Step 2: Press [ENTER]. y = 39.488 2.2.1: Calculating Sine, Cosine, and Tangent

  25. Guided Practice: Example 4, continued On a TI-Nspire: Check to make sure your calculator is in degree mode first. Refer to the directions in the previous example. Step 1: Press [trig][17][÷][cos][64.5]. Step 2: Press [enter]. y = 39.488 The measure of AB = 39.488. 2.2.1: Calculating Sine, Cosine, and Tangent

  26. Guided Practice: Example 4, continued Use the Pythagorean Theorem to check your trigonometry calculations. AC = 35.641 and AB = 39.488. 2.2.1: Calculating Sine, Cosine, and Tangent

  27. Guided Practice: Example 4, continued Find the value of ∠A. 2.2.1: Calculating Sine, Cosine, and Tangent

  28. Guided Practice: Example 4, continued Using trigonometry, you could choose any of the three functions since you have solved for all three side lengths. In an attempt to be as precise as possible, let’s choose the given side length and one of the approximate side lengths. 2.2.1: Calculating Sine, Cosine, and Tangent

  29. Guided Practice: Example 4, continued Use the inverse trigonometric function since you are solving for an angle measure. 2.2.1: Calculating Sine, Cosine, and Tangent

  30. Guided Practice: Example 4, continued On a TI-83/84: Step 1: Press [2ND][SIN][17][÷][39.488][)]. Step 2: Press [ENTER]. z = 25.500° On a TI-Nspire: Step 1: Press [trig][sin][17][÷][39.488]. Step 2: Press [enter]. z = 25.500° 2.2.1: Calculating Sine, Cosine, and Tangent

  31. Guided Practice: Example 4, continued Check your angle measure by using the Triangle Sum Theorem. ∠A is 25.5° ✔ 2.2.1: Calculating Sine, Cosine, and Tangent

  32. Guided Practice: Example 4, continued 2.2.1: Calculating Sine, Cosine, and Tangent

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