Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Solving Problems with Trigonometry Connections Have you ever . . . • Modeled a problem using a right triangle? • Had to find the height of a flagpole or column? • Wondered how far away a helicopter was? Trigonometry can be used to solve real-world situations involving triangles. For instance, a point on the ground, the top of a flagpole, and the bottom of a flagpole form a right triangle. Suppose you know the height of the flagpole and the angle of elevation from the bottom to the top of the flagpole. How can you use trigonometry to find the distance from the point on the ground to the bottom of the flagpole? Trigonometry is the study of the relationships between the sides and angles of a triangle. It can be used to find an unknown side or angle of a right triangle. A trigonometric ratio is a fraction formed by two sides of a triangle. Three important trigonometric ratios are sine (sin), cosine (cos) and tangent (tan). To find these ratios in a right triangle, start with one of the acute angles, such as the one labeled using the uppercase Greek letter theta (H) in the diagram. Notice which sides are the hypotenuse, opposite, and adjacent sides and how to calculate sine, cosine, and tangent. You can use these definitions to find unknown side lengths of a right triangle. sin H = opposite hypotenuse adjacent hypotenuse tan H = opposite adjacent H opposite cos H = se u oten p y h adjacent 1 Essential Math Skills Learn It! Using Trigonometry to Find a Distance Here’s an example of a problem using trigonometry. In the figure below, two side lengths are given, an angle measure is given, and the side labeled x is unknown. What if you want to find the value of x? sin(31) = x 7 cos(31) = 6 7 tan(31) = x 6 Math Tip You can use your calculator to find the sine, cosine, or tangent of an angle. Press the sin, cos, or tan button, then enter the angle measurement, and then press enter. Remember: Every right triangle has two acute angles and one right angle. The longest side of every right triangle is the hypotenuse. The other two sides are called legs. 7 x Math Tip 31c 6 You can write an equation to find the value of x. Two of the trigonometric ratios have the length x you want to find, so either one will do. Let’s use sin(31) = x . To find the value of x, you can multiply both sides by 7. 7 sin(31) = x $ 7sin(31) = x 7 So, x is equal to 7sin(31). You can use your calculator to find x as a decimal. Rounded to the nearest hundredth, sin(31) = 0.52, which you can use to find x. x = 7sin(31) = 7(0.52) = 3.64 When a right triangle represents a real-world situation, trigonometric ratios can help you to answer questions about missing information. A flagpole has a height of 35 feet. From a point on the ground, the angle of elevation from the bottom of the flagpole to the top of the flagpole is 52°. What is the distance from the point to the bottom of the flagpole? Draw and Label a Diagram to Visualize the Problem Drawing a diagram helps you to see how a right triangle models the problem. Use x to represent what you want to find. ? 2 1. On a separate sheet of paper, draw and label a right triangle to represent the problem. Solving Problems with Trigonometry Draw a right triangle so that the bottom leg represents the ground, and the other leg represents the height of the flagpole. Use x for the distance you want to find. 35 52c x Use the Diagram to Write a Trigonometric Ratio Using a trigonometric ratio will help you to find the missing information. ? 2. What trigonometric ratio can you use to help you find x? The sides opposite and adjacent to the 52° angle are the known side and the side you want to find. The only trigonometric ratio that uses opposite and adjacent is tangent: tan(52) = 35 x Solve the Equation and Write the Solution Solve the equation by getting the variable alone on one side of the equation. ? 3.Find x and then write the solution. Solve the equation by multiplying both sides by x and then dividing by tan(52) to get x alone on one side of the equation. tan(52) = 35 x $ x tan(52) = 35 $ x = 35 $ x = 35 tan(52) 1.28 $ x = 27.34 3 Essential Math Skills e ic Pract It! Answer the following questions. Round all decimal answers to the nearest hundredth. 1. You are standing on the ground 550 yards from the bottom of a cliff. From where you are standing, the angle of elevation from the bottom of the cliff to the top of the cliff is 8°. sin H = opposite hypotenuse cos H = adjacent hypotenuse tan H = opposite adjacent a. Draw a right triangle to represent this situation. b. Write a trigonometric ratio that can be used to find the height x of the cliff. c. Solve your ratio to find the height, in yards, of the cliff. 2. A helicopter is hovering 1,625 feet above the ground without moving. The angle of elevation from the landing pad to the helicopter is 22°. 4 a. Draw a right triangle to represent this situation. b. Write a trigonometric ratio that can be used to find the distance x from the helicopter to the landing pad. c. Solve your ratio to find the distance in feet from the helicopter to the landing pad. Solving Problems with Trigonometry 3. A 30-meter steel cable is connected at the top of a radio tower to a point on the ground. From this point, the angle of elevation from the bottom of the tower to the top of the tower is 48°. a. Draw a right triangle to represent this situation. b. Write a trigonometric ratio that can be used to find the distance x from the point to the bottom of the tower. c. Solve your ratio to find the distance in meters from the point on the ground to the bottom of the tower. 4. Jonas wants to find the value of x this the right triangle. 70c x Build Your Math Skills 16 You can use other facts about triangles along with trigonometric ratios. a. Write a trigonometric ratio that Jonas can use to find x. b. Solve your ratio to find x. Remember that the Pythagorean theorem can help you calculate the length of sides. Understanding that the sum of the angles in a trianlge equals 180° can help you calculate unknown angles. 5 Essential Math Skills Math Tip 5. A scuba diver is 14.5 meters below sea level. The angle of decline from a boat to the diver is 30°. An angle of elevation is an angle formed from the horizontal line up to a point. a. Draw a right triangle to represent this situation. An angle of decline is an angle formed from the horizontal line down to a point. b. Write a trigonometric ratio that can be used to find the distance x from the boat to the diver. c. Solve your ratio to find the distance in meters from the point on the ground to the bottom of the tower. 6. In a right triangle, one of the angles is 70° and the side opposite the 70° angle is 14 inches. 6 a. Draw a right triangle to represent this situation. b. Write a trigonometric ratio that can be used to find the side x that is adjacent to the 70° angle. c. Solve your ratio to show that the length of the adjacent side is 14 inches. tan(70) Solving Problems with Trigonometry Check Your Skills Use your knowledge of right triangles and trigonometry to answer the following questions. 1. What is the value of cos(62) rounded to the nearest hundredth? a. 0.88 b. 0.47 c. 1.88 d. 0.62 2. For the figure below, which answer is equal to cos H? H 17 8 15 3. 8 a. 17 15 b. 8 17 c. 8 15 d. 17 A lamp post casts a shadow 6.8 meters long. From the end of the shadow, the angle of elevation from the bottom to the top of the lamp post is 51°. How tall is the lamp post, rounded to the nearest tenth of a meter? a. 8.4 m b. 5.9 m c. 4.3 m d. 6.8 m 7 Essential Math Skills 4. Select all equations that are true for the figure below. 25 7 H 24 25 o sin H = 24 24 o cos H = 25 7 o tan H = 25 24 o sin H = 7 7 o cos H = 25 7 o tan H = 24 5. A wire is connected from the top of a tower to a point on the ground that is 62 feet from the bottom of the tower. From this point, the angle of elevation from the bottom of the tower to the top of the tower is 75°. What is the length of the wire? 62 a. ft tan(75) b. 62 cos(75) ft c. 62 tan(75) ft 62 d. ft cos(75) 6. For the figure below, what is the value of sin H? 3 a. 5 4 b. 5 5 c. 3 5 3 d. 8 H 3 8 Remember the Concept Trigonometry can be used to solve problems involving right triangles. You can find an unknown side length using trigonometry when you know the measure of one of the acute angles and the length of another side. Answers and Explanations Solving Problems with Trigonometry Using Trigonometry to Find a Distance 2c.4337.88 feet Solve the equation. sin(22) = 1,625 x Practice It! First, multiply by x. 1a. x sin(22) = 1,625 Then, divide by sin(22). 8c x = 1,625 sin(22) 550 The distance 550 is the distance along the ground to the bottom of the cliff, and the smallest angle is 8° (the angle of elevation to the top of the cliff). x 1b. tan(8) = 550 Use x to represent the height of the cliff. Use the tangent ratio because the height x of the cliff is opposite the 8° angle, and the distance from you to the bottom of the cliff (550 yards) is the adjacent leg. Using a calculator, divide 1,625 by sin(22). x = 4337.88 3a. 30 1c.77.30 yards Solve the equation. tan(8) = 48c x 550 Multiply both sides by 550. 550 tan(8) = x Using a calculator, multiply 550 by tan(8). x = 77.30 2a. 1,625 22c The distance 1,625 is the height above the ground, so this distance and the ground make the legs of the triangle. The elevation of 22° is the angle from the ground toward the helicopter. 2b. sin(22) = 1,625 x Use x to represent the distance from the landing pad to the helicopter (the hypotenuse). Use the sine ratio because the height (1,625 ft) of the helicopter is opposite the 22° angle, and the distance x from the helicopter to the landing pad is the hypotenuse. The distance 30 would be the hypotenuse of the triangle, because it’s the distance from the point to the top of the tower. The angle 48° is the angle from the point, looking up to the top of the tower. x 3b. cos(48) = 30 Use x to represent the distance from the point to the bottom of the tower. Use the cosine ratio because the distance x from the point to the bottom of the tower is the adjacent side, and the length of the cable (30 m) is the hypotenuse. 3c.20.07 meters Solve the equation. cos(48) = x 30 Multiply both sides by 30: 30 cos(48) = x Using a calculator, multiply 30 by cos(48). x = 20.07 x 16 Use the sine ratio because the side opposite the 70° angle is x, and the hypotenuse is 16. 4a. sin(70) = i Essential Math Skills 4b.15.04 6a. Solve the equation. sin(70) = x 16 Multiply both sides by 16: 16 sin(70) = x Using a calculator, multiply 16 by sin(70). 14 x = 15.04 5a. boat 70c 30c 14.5 diver The distance 14.5 is the leg down from the surface of the water to the diver. The angle 30° is the angle from the boat, looking down at the diver. 5b. sin(30) = 14.5 x Use x to represent the distance from the boat to the diver. Use the sine ratio because the depth of the diver (14.5 m) is opposite the 30° angle, and the distance x from the boat to the diver is the hypotenuse of the right triangle. 5c.29.00 meters Solve the equation. sin(30) = 14.5 x Use x to represent the adjacent side. Use the tangent ratio because the opposite side (14 in.) is given, and x is the adjacent side. 6c.Divide by x and then divide by tan(70) to show that x is equal to this ratio: 14 tan(70) tan(70) = 14 x Multiply both sides by x. x tan(70) = 14 Divide both sides by tan(70). x= 14 tan(70) Multiply both sides by x. Check Your Skills x sin(30) = 14.5 1. b. 0.47 Divide both sides by sin(30). Use a calculator. Press the cos button, then 62, and then press enter. Round the result to the nearest hundredth (2 decimal places). 8 2. a. 17 x = 14.5 sin(30) Using a calculator, divide 14.5 by sin(30). x = 29.00 ii The triangle may be turned a different way, but the leg that is length 14 will be opposite the angle that is 70°. 6b. tan(70) = 14 x The cosine ratio is the side adjacent to H (which is 8) divided by the hypotenuse (which is 17). Answers and Explanations 3. a. 8.4 m Draw a diagram using x for the height of the lamp post. 62 ft cos(75) Draw a diagram and let x represent the length of the wire. 5. d. x x 51c 6.8 m The sides opposite and adjacent to the 51° angle are the side you want to find and the known side, which means you use the tangent ratio: tan(51) = x 6.8 To find x, multiply both sides by 6.8: 6.8 tan(51) = x Using a calculator, multiply 6.8 by tan(51). x = 8.4 4. cos H = 24 25 tan H = 7 24 The hypotenuse is 25, the opposite side is 7, and the adjacent side is 24. This triangle has the following trigometric ratios: sin H = 7 25 cos H = 24 25 tan H = 7 24 75c 62 ft Use the cosine ratio because 62 is the adjacent side and x is the hypotenuse: cos(75) = 62 x Multiply both sides by x: x cos(75) = 62 Sove for x by dividing both sides by cos(75): x= 62 cos(75) 4 6. b. 5 To find sin H, you need to find the opposite side, which is unknown. Use the Pythagorean theorem using a for the length of the unknown side: a2 + 32 = 52 Calculate the exponents: a2 + 9 = 25 Subtract 9 from each side: a2 = 16 Take the square root to find the value of a. The value must be positive, since it’s the measurement of a length. a = 16 = 4 The opposite side is 4, and the hypotenuse is 5. Therefore, sin H = 4 . 5 iii