Practice A. Solving Right Triangles. sin. cos A 5. tan 2
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1 Name Date Class Solving Right Triangles In Exercises 1 3, fill in the blanks to complete the description of the inverse trigonometric ratios. 1. If sin A = x, then sin 1 x =. 2. If cos A =, then cos 1 x = m A. 3. If tan A = x, then = m A. Use the given trigonometric ratio to determine whether 1 or 2 is A in each exercise. 4. sin A = 4. cos A = 4 6. tan A = sin A = 3 8. cos = 3 A 9. tan A = 4 3 Use a calculator to find each angle measure to the nearest degree. 10. sin 1 (0.33) 11. cos 1 (0.47) 12. tan 1 (1.21) 13. sin cos Use a calculator and inverse trigonometric ratios to find the unknown side lengths and angle measures. Round lengths to the nearest hundredth and angle measures to the nearest degree tan AC = DE = GH = m B = EF = m H = m C = m D = m I =!XYZ has vertices X(6, 6), Y(6, 3), and Z(1, 3). Complete Exercises to find the side lengths to the nearest hundredth and the angle measures to the nearest degree. 19. Plot the points and draw!xyz. 20. Tell which angle is the right angle. 21. Find XY and YZ from the graph. Use the Pythagorean Theorem to find XZ. XY = YZ = XZ =
2 Name Date Class Solving Right Triangles Use the given trigonometric ratio to determine which angle of the triangle is A. 1. sin A = 8 2. cos A = 1 3. tan A = sin A = 1. cos A = 8 6. tan A = 8 1 Use a calculator to find each angle measure to the nearest degree. 7. sin 1 (0.82) sin cos cos 1 (0.23) 12. Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree tan 1 (.03) tan For each triangle, find all three side lengths to the nearest hundredth and all three angle measures to the nearest degree. 19. B( 2, 4), C(3, 3), D( 2, 3) _ 20. L( 1, 6), M(1, 6), N( 1, 1) _ 21. X( 4, ), Y( 3, ), Z( 3, 4) _
3 Name Date Class Measure of T Length US Tan T Length TS Sec T ft ft inches inches km The length of US is close in value to tan T, and the length of TS is close in value to sec T. Problem Solving ft cm cm in 2. C 6. G 7. A 8. F Reading Strategies 1. Answers will vary. 2. Both ratios have Hypotenuse in the denominator. 3. hypotenuse 4. a ; 0.38 b ; 0.92 c ; 0.42 SOLVING RIGHT TRIANGLES 1. m A 2. x 3. tan 1 x yd Y 21. 3; ; AB = 7.74 in.; m A = 7 ; m B = EF = 2.73 m; m D = 6 ; m F = 2 1. GH = 7.64 ft; GI = 7.91; m I = KL = 2.71 yd; JK = 2.84 yd; m K =. QP = cm; m Q = 42 ; m R = ST = 3.8 yd; m S = 12 ; m T = BC = 8.60; BD = 7; CD = ; m B = 36 ; m C = 4 ; m D = LM = 2; LN = 7; MN = 7.28; m L = 90 ; m M = 74 ; m N = XY = 1; XZ = 1.41; YZ = 1; m X = 4 ; m Y = 90 ; m Z = 4 Practice C ; ; ; ; 74. Yes, a 20% grade in the United States is equal to a 20% grade elsewhere. A 20% grade in the
4 Name Date Class Angles of Elevation and Depression Marco breeds and trains homing pigeons on the roof of his building. Classify each angle as an angle of elevation or an angle of depression To attract customers to his car dealership, Frank tethers a large red balloon to the ground. In Exercises 7, give answers in feet and inches to the nearest inch. (Note: Assume the cord that attaches to the balloon makes a straight segment.). The sun is directly overhead. The shadow of the balloon falls 14 feet 6 inches from the tether. Frank sights an angle of elevation of 67. Find the height of the balloon. 6. Find the length of the cord that tethers the balloon. 7. The wind picks up and the angle of elevation changes to 9. Find the height of the balloon. Lindsey shouts down to Pete from her third-story window. 8. Lindsey is 9.2 meters up, and the angle of depression from Lindsey to Pete is 79. Find the distance from Pete to the base of the building to the nearest tenth of a meter. 9. To see Lindsey better, Pete walks out into the street so he is 4.3 meters from the base of the building. Find the angle of depression from Lindsey to Pete to the nearest degree. 10. Mr. Shea lives in Lindsey s building. While Pete is still out in the street, Mr. Shea leans out his window to tell Lindsey and Pete to stop all the shouting. The angle of elevation from Pete to Mr. Shea is 72. Tell whether Mr. Shea lives above or below Lindsey. _
5 Name Date Class Angles of Elevation and Depression In Exercises 1 and 2, fill in the blanks to complete the definitions. 1. An angle of elevation is the angle formed by a line and a line of sight to a point the line. 2. An angle of is the angle formed by a horizontal line and a line of sight to a point the line. Ben is on the diving board at the neighborhood pool. Jenna is in the pool, and a lifeguard sits at her station on the opposite end of the pool. Classify each angle as an angle of elevation or an angle of depression Lisa sees a bird s nest high in a tree. She decides to use trigonometry to estimate how high the nest is. 7. Lisa walks 1 feet from the base of the tree. She measures an angle of elevation from the ground to the nest of 62. Find how high the nest is above the ground, to the nearest foot. 8. Lisa spots the mother bird on a branch above the nest. She measures an angle of elevation to the bird of 67. Find how high the mother bird is above the ground, to the nearest foot. Zelda, a trapeze artist, stands on a 10-meter-high platform. 9. Zelda measures a 40 angle of depression to the base of the other platform. Find the distance between the bases of the platforms to the nearest tenth of a meter 10. Zelda s partner, Zev, is on the ground doing a safety check on the net. Zelda measures a 79 angle of depression to Zev. Find the distance to the nearest tenth of a meter from Zev to the base of Zelda s platform.
6 Name Date Class United States means a rise in elevation of 20 feet over 100 horizontal feet. A 20% grade elsewhere means a rise in elevation of 20 meters over 100 meters. 20 ft But a grade is a ratio: = 20 m. 100 ft 100 m The units cancel out, and either way a 20% grade simplifies to 1, or an angle with the horizontal that measures about EG = 0.61 m; FG = 0.4 m; m E = 7 7. KM = 6.13 mm; m K = 61 ; m L = BC = 3.74 ft; m B = 83 ; m D = TV = 8.43 in.; UV = in.; m T = IJ =.32 yd; m H = 90 ; m I = RS = mm; ST = mm; m R = 60 Reteach AB 12.2 ft; AC 7.4 ft; m B = FH 9.12 mi; m F 26 ; m H QR km; QS 2.04 km; m Q = WX cm; m X 9 ; m Y MP = MN = 6; PN 8.49; m M = 90 ; m P = 4 ; m N = KL = 7; LJ = 3; JK 7.62; m L = 90 ; m J 67 ; m K 23 Challenge sin E = ; cos E = 2 ; m E. sin M = 1 2 ; tan M = 3 3 ; m M = cos R = 7. sin K = 2 2 Problem Solving ; tan R = 1; m R = ; cos K = 1 ; m K = to ft. A 6. G 7. D 8. F Reading Strategies 1. finding the measures of all unknown sides and angles of the triangle 9 2. m B = tan sin 24 = ; 23 ; 12 AB ANGLES OF ELEVATION AND DEPRESSION 1. horizontal; above 2. depression; below 3. angle of depression 4. angle of elevation. angle of depression 6. angle of elevation feet 8. 3 feet meters meters 1. angle of elevation 2. angle of depression 3. angle of depression 4. angle of elevation. 34 ft 2 in ft 1 in ft 10 in m Mr. Shea lives above Lindsey.
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