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Geometry Chapter 5 Lesson 8 – Applying Special Right Triangles Learning Target: (LT-7) Solve problems involving special right triangles. Ex1: Find missing sides in 45º-45º-90º Triangles Find the value of x. Give your answer in simplest radical form. a. b. d. e. c. f. Page 1 Ex2: Find missing sides in 30º- 60º - 90º Triangles Find the values of x and y. Give your answers in simplest radical form. a. b. c. d. e. f. Page 2 Chapter 8 Trigonometric Ratios Lesson 8.1 Similarity in Right Triangles Learning Target: (LT-1) Apply similarity relationships, including the geometric mean, in right triangles to solve problems. Geometric Mean: 1. Write a similarity statement comparing the three triangles. 2. Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. A. 4 and 25 B. 5 and 30 C. 8 and 9 3. Find x, y, and z. 4. To estimate the height of a Douglas fir, Jan positions herself so that her line of sight to the top and bottom of the tree form a 90°angle. Her eyes are about 1.6m above the ground, and she is standing 7.8m from the tree. What is the height of the tree to the nearest meter? Page 3 Lesson 8.2 Trigonometric Ratios Learning Target: (LT-2) Use trigonometric ratios of right triangles to find unknown lengths. • trigonometric ratio: By the AA Similarity Theorem, a right triangle with a given acute angle is similar to every other right triangle with that same acute angle measure. Examples: 1. Write each trigonometric ratio as a fraction and a decimal 2. Use a special right triangle to write cos 30° as a fraction. rounded to the nearest hundredth. A. sin J B. cos J C. tan K 3. Use a special right triangle to write sin 60° as a fraction. 4. Use your calculator to find each trigonometric ratio, Round to the nearest hundredth. A. sin 52° B. cos 19° C. tan 65° Page 4 4. Find each length. Round to the nearest hundredth. A. BC 5. The Pilatusbahn in Switzerland is the world's steepest cog railway. Its steepest section makes an angle of about 25.6° with the horizontal and rises about 0.9 km. To the nearest hundredth of a kilometer, how long is this section of the railway track? B. QR C. FD Ex6. Use special triangles to complete the chart. Answers must be exact values in simplified radical form. 30º 45º 60º sin cos tan Page 5 Lesson 8.3 – Solving Right Triangles Learning Target: (LT-3) Use inverse trigonometric ratios of right triangles to find unknown angles. Solving a triangle – Examples: 1. Use the trigonometric ratio cosA = 24 25 to determine which angle of the triangle is ∠A. 2. Use your calculator to find each angle measure to the nearest degree. A. cos-1 (0.87) B. sin-1 (0.85) C. tan-1 (0.71) 3. Solve the triangle. Round lengths to the nearest hundredth and angle measures to the nearest degree. 4. A highway sign warns that a section of road ahead has a 7% grade. To the nearest degree, what angle does the road make with a horizontal line? 5. The coordinates of the vertices of ΔPQR are P(-3, 3), Q(2, 3), and R(-3, -4). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. Page 6 Lesson 8-4 Angles of Elevation and Depression Learning Target: (LT-4) Solve problems involving angles of depression and angles of elevation. Angle of elevation – Angle of depression – Examples: 1. Classify each angle as an angle of elevation or an angle of depression. 2. The seattle Space Needle casts a 67ft. shadow. If the angle of elevation from the tip of the shadow to the top of the Space Needle is 70°, how tall is the Space Needle? Round to the nearest meter. 3. An ice climber stands at the edge of a crevasse that is 115ft. wide. The angle of depression from the edge where she stands to the bottom of the opposite side is 52°. How deep is the crevasse at this point? Round to the nearest foot. 4. What if…? Suppose the ranger sees another fire and the angle of depression to the fire is 3°. What is the horizontal distance to this fire? Round to the nearest foot. 5. 4. An observer in a lighthouse is 69ft. above the water. He sights two boats in the water directly in front of him. The angle of depression to the nearest boat is 48°. The angle of depression to the other boat is 22°. What is the distance between the two boats? Round to the nearest foot. Page 7 Lesson 8.5 – Law of Sines and Cosines Learning Target: (LT-5a) Use the Law of Sines to solve problems. Examples Using The Law of Sines #1. Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. A. tan 103° B. cos 165° C. sin 93° #2. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. A. FG B. Page 8 m ∠Q Learning Target: (LT-5b) Use the Law of Cosines to solve problems. Examples Using The Law of Cosines Ex#3: Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. B. m ∠T A. XZ Ex#4: A sailing club has planned a triangular racecourse, as shown in the diagram. How long is the leg of the race along BC ? How many degrees must competitors turn at point C? Round the length to the nearest tenth and the angle measure to the nearest degree. Page 9 Lesson 8.6 – Vectors Learning Target: (LT-6) Use vectors to solve problems. Vocabulary • vector – • component form – • magnitude- • direction – • equal vectors – • parallel vectors – • resultant vector – Page 10 Examples: 2. Draw the vector 〈−1, 5〉 on a coordinate plane. Find its magnitude to the nearest tenth. 1. Write each vector in component form. A. HG B. MN with M(-8, 1) and N(2, -7). 3, The force exerted by a skier is given by the vector 〈1, 4〉 . Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. 4. Identify each of the following. A equal vectors. B. parallel vectors. Page 11 Lesson Problems 5.8 p. 360 #1-8, 17, 18 & Special Triangles Worksheet 8.1 p. 521 #15-39, 42, 47-49, 54, 57-59 8.2 p. 529 #22-50, 53, 54, 62, 68, 70, 84-86 8.3 p. 538 #21-29, 33, 34, 36, 39-44, 48, 52, 65-67 8.4 p. 547 #10-14, 17-22, 23 8.5 p. 555 Law of Sines #23-28, 30, 39, 47, 49, 60, 61, 68-74 8.5 p. 555 Law of Cosines #32-35, 38, 40, 44-46, 51-54, 59 8.6 p. 564 #18-35, 38, 39, 46, 47, 53, 54, 60, 63 Page 12