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Name: ________________________ Class: ___________________ Date: __________ ID: A Trigonometry Partner Assignment Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Evaluate tan 4°, to four decimal places. a. 1.6949 b. 0.9976 c. 0.6993 d. 0.0699 5. A school soccer field measures 45 m by 65 m. To get home more quickly, you decide to walk along the diagonal of the field. What is the angle of your path, with respect to the 45-m side, to the nearest degree? a. 55° b. 34° c. 2° d. 1° 2. In the triangle, BC = 12 cm and tan C = 0.583. What is the length of the hypotenuse, to the nearest tenth of a centimetre? Use the diagram to answer the following question(s). a. b. c. d. 14.2 cm 13.9 cm 7.0 cm 5.1 cm 6. Determine the length of x, to the nearest tenth of a metre. a. 5.7 m b. 6.3 m c. 7.0 m d. 8.2 m 3. Evaluate tan 45°. a. –1 b. 0 c. 1 d. undefined 7. Determine the length of y, to the nearest tenth of a metre. a. 5.7 m b. 6.3 m c. 7.0 m d. 8.2 m 4. Determine the measure of ∠C, to the nearest degree. a. b. c. d. 19° 20° 21° 22° 1 Name: ________________________ ID: A 11. Write the cosine ratio of ∠A. 8. Choose the correct formula for the cosine ratio of ∠A. length of side opposite ∠A a. cos A = length of side adjacent to ∠A length of hypotenuse b. cos A = length of side adjacent to ∠A length of side opposite ∠A c. cos A = length of hypotenuse length of side adjacent to ∠A d. cos A = length of hypotenuse a. b. 9. Determine the value of sin 28°, to four decimal places. a. 0.4695 b. 0.5317 c. 0.8829 d. 0.9709 c. d. 12. In the triangle, AC = 26 cm and AB = 10 cm. Determine the sine ratio of ∠A, to the nearest thousandth. 10. In ABC, AC = 8 cm and BC = 11 cm. Determine the sine ratio of ∠B, rounded to the nearest thousandth. a. b. c. d. AB AC BC AB AC BC BC AC a. b. c. d. 0.588 0.728 0.809 1.375 2 0.385 0.417 0.923 1.833 Name: ________________________ ID: A 13. In ABC, BC = 48 cm and sin A = 0.96. Determine the length of AC, to the nearest centimetre. a. b. c. d. Use the diagram to answer the following question(s). Kelly is flying a kite in a field. He lets out 40 m of his kite string, which makes an angle of 72° with the ground. 14 cm 24 cm 25 cm 50 cm 14. In the triangle, BC = 11 cm and cos B = 0.809. Determine the length of AC, to the nearest centimetre. a. b. c. d. 17. Determine the height of the kite above the ground, to the nearest metre. a. 12 m b. 38 m c. 42 m d. 129 m 18. Suppose the sun is shining directly above the kite. How far is the kite’s shadow from Kelly, to the nearest metre? a. 12 m b. 38 m c. 42 m d. 129 m 8 cm 9 cm 10 cm 13 cm 15. Determine the value of cos 0°. a. –1 b. 0 c. 1 d. undefined 19. An Airbus A320 aircraft is cruising at an altitude of 10 000 m. The aircraft is flying in a straight line away from Monica, who is standing on the ground. Monica observes that the angle of elevation of the aircraft changes from 70° to 33° in one minute. What is the cruising speed of the aircraft, to the nearest kilometre per hour? a. 463 km/h b. 485 km/h c. 665 km/h d. 705 km/h 16. In TUV, UV = 8 m, ∠U = 90°, and ∠T = 38°. Determine the measure of ∠V, to the nearest degree a. 42° b. 52° c. 90° d. 128° 3 Name: ________________________ ID: A 8 . 11 ABC, to the nearest square 20. In the triangle, BC = 11 cm and tan B = Determine the area of centimetre. a. b. c. d. 121 cm2 88 cm2 64 cm2 44 cm2 Completion Complete each statement. 1. The hypotenuse is the side _________________________ the right angle in a right triangle. 6. The angle between the horizontal and the line of sight looking up to an object is called the angle of _________________________ . 2. The tangent ratio of ∠A can be written as _________________________ . 7. The angle of depression is the angle between the horizontal and the line of sight looking _________________________ to an object. 3. The sine ratio of ∠A compares the length of the _________________________ side to the length of the hypotenuse. 8. The _________________________ can be used to determine the length of one side of a right triangle given the lengths of the other two sides. 4. The cosine ratio of ∠A compares the length of the _________________________ side to the length of the hypotenuse. 9. In a right triangle, the _________________________ side is the side that forms one of the arms of the angle being considered but is not the hypotenuse. 5. The angle of elevation is the angle between the horizontal and the line of sight looking _________________________ to an object. 10. The angle between the horizontal and the line of sight looking down to an object is called the angle of _________________________ . 4 Name: ________________________ ID: A Matching Match the correct side or angle of PQR to each of the following definitions, descriptions, or expressions. A term may be used more than once or not at all. a. b. c. p p q p r f. q r g. –1 h. 0 d. Pythagorean theorem i. e. q j. 1 pq 2 1 1. the side opposite ∠Q 3. r 2 = q 2 + p 2 2. the side adjacent to ∠Q 4. sin Q Match each term to the correct definition, description, or expression. A term may be used more than once or not at all. a. angle of elevation e. Pythagorean theorem b. angle of depression f. sine ratio c. cosine ratio g. tangent ratio d. hypotenuse h. trigonometry length of side opposite an acute angle length of side adjacent to an acute angle 5. in a right triangle with side lengths a, b, and c, where c is the hypotenuse, a 2 + b 2 = c 2 8. 6. the study of angles and triangles 9. the side opposite the right angle in a right triangle 7. length of side opposite an acute angle length of hypotenuse 10. 5 length of side adjacent to an acute angle length of hypotenuse Name: ________________________ ID: A Short Answer 1. Juanita is helping build a wheelchair ramp at her community centre. The ramp must reach a vertical distance of 1.5 m. If the ramp must make an angle of 6° with the ground, how long will the ramp be? Answer to the nearest tenth of a metre. 3. In right triangle ABC, the hypotenuse, AB, is 15 cm and ∠B is 25°. How long is AC, to the nearest centimetre? 2. A telephone pole is secured with a guy wire as shown in the diagram. The guy wire makes an angle of 75° with the ground and is secured to the ground 6 m from the bottom of the pole. Determine the height of the telephone pole, to the nearest tenth of a metre. 4. In XYZ, XY and XZ have equal lengths of 6 cm. YZ is 10 cm. Determine the measure of ∠X, to the nearest degree. 5. In the diagram, label the hypotenuse, the opposite side, and the adjacent side relative to ∠C. Problem 1. The percent grade of a road is the ratio of the vertical rise of the road to the horizontal distance, expressed as a percent. A road in the mountains has a vertical rise of 40 m over 1 km of horizontal distance. a) Determine the percent grade of the road, to the nearest percent. b) What is the angle of elevation of the road, to the nearest degree? 2. Myriam is participating in a water-skiing competition. She goes over a water-ski ramp that is 4.0 m long. When she leaves the ramp, she is 1.7 m above the surface of the water. a) What is the horizontal length of the ramp along the surface of the water, to the nearest tenth of a metre? b) Determine the angle of elevation of the ramp, to the nearest degree. 6 ID: A Trigonometry Partner Assignment Answer Section MULTIPLE CHOICE 1. ANS: NAT: 2. ANS: NAT: KEY: 3. ANS: NAT: 4. ANS: NAT: KEY: 5. ANS: NAT: KEY: 6. ANS: NAT: KEY: 7. ANS: NAT: KEY: 8. ANS: NAT: 9. ANS: NAT: 10. ANS: NAT: KEY: 11. ANS: NAT: KEY: 12. ANS: NAT: KEY: 13. ANS: NAT: KEY: 14. ANS: NAT: KEY: 15. ANS: NAT: 16. ANS: NAT: D PTS: 1 DIF: B OBJ: Section 3.1 M4 TOP: The Tangent Ratio KEY: tangent ratio | calculate a tangent ratio B PTS: 1 DIF: C OBJ: Section 3.1 M4 TOP: The Tangent Ratio tangent ratio | determine a distance using trigonometry | Pythagorean theorem | hypotenuse | right triangle C PTS: 1 DIF: A OBJ: Section 3.1 M4 TOP: The Tangent Ratio KEY: tangent ratio | calculate a tangent ratio C PTS: 1 DIF: B OBJ: Section 3.1 M4 TOP: The Tangent Ratio tangent ratio | determine an angle measure | right triangle A PTS: 1 DIF: C OBJ: Section 3.3 M4 TOP: Solving Right Triangles tangent ratio | determine an angle measure C PTS: 1 DIF: C OBJ: Section 3.1 M4 TOP: The Tangent Ratio tangent ratio | determine a distance using trigonometry B PTS: 1 DIF: C OBJ: Section 3.1 M4 TOP: The Tangent Ratio tangent ratio | determine a distance using trigonometry D PTS: 1 DIF: A OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios KEY: cosine ratio | define the cosine ratio A PTS: 1 DIF: A OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios KEY: sine ratio | calculate a sine ratio A PTS: 1 DIF: B OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios sine ratio | calculate a sine ratio | right triangle A PTS: 1 DIF: A OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios cosine ratio | write a cosine ratio | right triangle C PTS: 1 DIF: B OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios sine ratio | calculate a sine ratio | right triangle A PTS: 1 DIF: B OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios sine ratio | determine a distance using trigonometry | right triangle A PTS: 1 DIF: B OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios cosine ratio | determine a distance using trigonometry | right triangle C PTS: 1 DIF: A OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios KEY: cosine ratio | calculate a cosine ratio B PTS: 1 DIF: A OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios KEY: sine ratio | determine an angle measure 1 ID: A 17. ANS: NAT: KEY: 18. ANS: NAT: KEY: 19. ANS: NAT: 20. ANS: NAT: B PTS: 1 DIF: B OBJ: Section 3.3 M4 TOP: Solving Right Triangles sine ratio | determine a distance using trigonometry | solve a right triangle A PTS: 1 DIF: C OBJ: Section 3.3 M4 TOP: Solving Right Triangles cosine ratio | determine a distance using trigonometry | solve a right triangle D PTS: 1 DIF: D OBJ: Section 3.3 M4 TOP: Solving Right Triangles KEY: sine ratio | solve a right triangle D PTS: 1 DIF: C OBJ: Section 3.3 M4 TOP: Solving Right Triangles KEY: tangent ratio | right triangle | area COMPLETION 1. ANS: opposite PTS: 1 DIF: A TOP: The Tangent Ratio 2. ANS: Example: length of side opposite ∠A tanA = length of side adjacent to ∠A OBJ: Section 3.1 NAT: M4 KEY: right triangle | hypotenuse PTS: TOP: KEY: 3. ANS: 1 DIF: B OBJ: Section 3.1 NAT: M4 The Tangent Ratio tangent ratio | write a tangent ratio | primary trigonometric ratios opposite PTS: TOP: KEY: 4. ANS: 1 DIF: A OBJ: Section 3.2 NAT: M4 The Sine and Cosine Ratios sine ratio | define the sine ratio | opposite side | primary trigonometric ratios adjacent PTS: TOP: KEY: 5. ANS: 1 DIF: A OBJ: Section 3.2 NAT: M4 The Sine and Cosine Ratios cosine ratio | define the cosine ratio | adjacent side | primary trigonometric ratios up PTS: 1 DIF: A TOP: Solving Right Triangles 6. ANS: elevation OBJ: Section 3.3 NAT: M4 KEY: angle of elevation PTS: 1 DIF: B TOP: Solving Right Triangles 7. ANS: down OBJ: Section 3.3 NAT: M4 KEY: angle of elevation PTS: 1 DIF: A TOP: Solving Right Triangles OBJ: Section 3.3 NAT: M4 KEY: angle of depression 2 ID: A 8. ANS: Pythagorean theorem PTS: 1 DIF: A TOP: Solving Right Triangles 9. ANS: adjacent OBJ: Section 3.3 NAT: M4 KEY: Pythagorean theorem PTS: 1 DIF: B TOP: The Sine and Cosine Ratios 10. ANS: depression OBJ: Section 3.2 NAT: M4 KEY: adjacent side PTS: 1 DIF: A TOP: Solving Right Triangles OBJ: Section 3.3 NAT: M4 KEY: angle of depression MATCHING 1. ANS: NAT: 2. ANS: NAT: 3. ANS: NAT: 4. ANS: NAT: 5. ANS: NAT: 6. ANS: NAT: 7. ANS: NAT: 8. ANS: NAT: KEY: 9. ANS: NAT: 10. ANS: NAT: KEY: E M4 A M4 D M4 F M4 PTS: TOP: PTS: TOP: PTS: TOP: PTS: TOP: 1 DIF: A The Tangent Ratio 1 DIF: A The Tangent Ratio 1 DIF: A The Tangent Ratio 1 DIF: B The Sine and Cosine Ratios OBJ: KEY: OBJ: KEY: OBJ: KEY: OBJ: KEY: Section 3.1 label a right triangle | opposite side Section 3.1 label a right triangle | adjacent side Section 3.1 Pythagorean theorem Section 3.2 sine ratio | primary trigonometric ratios E PTS: 1 DIF: A M4 TOP: Solving Right Triangles H PTS: 1 DIF: A M4 TOP: The Tangent Ratio F PTS: 1 DIF: B M4 TOP: The Sine and Cosine Ratios G PTS: 1 DIF: B M4 TOP: The Tangent Ratio tangent ratio | primary trigonometric ratios D PTS: 1 DIF: A M4 TOP: The Tangent Ratio C PTS: 1 DIF: B M4 TOP: The Sine and Cosine Ratios cosine ratio | primary trigonometric ratios OBJ: KEY: OBJ: KEY: OBJ: KEY: OBJ: Section 3.3 Pythagorean theorem Section 3.1 trigonometry Section 3.2 sine ratio | primary trigonometric ratios Section 3.1 3 OBJ: Section 3.1 KEY: hypotenuse OBJ: Section 3.2 ID: A SHORT ANSWER 1. ANS: Let x represent the length of the ramp, in metres. vertical height of ramp sin6° = length of ramp sin6° = x= 1.5 x 1.5 sin6° x = 14.3502. . . The wheelchair ramp must be approximately 14.4 m long. PTS: 1 DIF: B OBJ: Section 3.1 NAT: M4 TOP: The Tangent Ratio KEY: sine ratio | determine a distance using trigonometry 2. ANS: Let h represent the height of the pole, in metres. height of pole tan75° = distance of wire from base tan75° = h 6 22.3923. . . = h The pole is approximately 22.4 m tall. PTS: 1 DIF: B OBJ: Section 3.1 NAT: M4 TOP: The Tangent Ratio KEY: tangent ratio | determine a distance using trigonometry | right triangle 3. ANS: side opposite ∠B sin B = hypotenuse sin 25° = AC 15 15(sin 25°) = AC 6.3393. . . = AC AC is approximately 6 cm long. PTS: 1 DIF: B OBJ: Section 3.2 NAT: M4 TOP: The Sine and Cosine Ratios KEY: sine ratio | determine a distance using trigonometry | right triangle 4 ID: A 4. ANS: cos Z = side adjacent to ∠Z hypotenuse cos Z = ZW XZ cos Z = 5 6 cos Z = 0.8333 Z = cos −1(0.8333) Z = 33.5573. . . ∠Z measures approximately 34°. ∠Y is equal to ∠Z. 180 = ∠X + ∠Y + ∠Z 180 = ∠X + 34 + 34 180 = ∠X + 68 112 = ∠X ∠X measures approximately 112°. PTS: 1 DIF: C TOP: The Sine and Cosine Ratios 5. ANS: PTS: 1 DIF: TOP: The Tangent Ratio A OBJ: Section 3.2 NAT: M4 KEY: cosine ratio | determine an angle measure | isosceles triangle OBJ: Section 3.1 NAT: M4 KEY: primary trigonometric ratios | label a right triangle 5 ID: A PROBLEM 1. ANS: a) 1 km = 1000 m ÊÁ ˆ˜ vertical rise ˜˜ × 100 percent grade = ÁÁÁ ÁË horizontal distance ˜˜¯ ÁÊ 40 ˜ˆ˜ ˜˜ × 100 percent grade = ÁÁÁÁ ˜ Ë 1000 ¯ percent grade = 0.04 × 100 percent grade = 4 The percent grade of the road is 4%. b) Let θ represent the angle of elevation, in degrees. vertical rise tan θ = horizontal distance tan θ = 40 1000 θ = tan−1 (0.04) θ = 2.2906. . . The angle of elevation of the road is approximately 2°. PTS: 1 DIF: D TOP: Solving Right Triangles OBJ: Section 3.3 NAT: M4 KEY: tangent ratio | angle of elevation 6 ID: A 2. ANS: a) Let x represent the length of the ramp along the surface of the water, in metres. Draw a diagram of the situation. x 2 + 1.7 2 = 4.0 2 x 2 = 16 − 2.89 x 2 = 13.11 x ≈ 3.62 The horizontal length of the ramp along the surface of the water is about 3.6 m. b) sinθ = 1.7 4.0 sinθ = 0.425 θ = 25.1507. . . θ ≈ 25 The angle of elevation of the ramp is about 25°. PTS: 1 DIF: B TOP: The Sine and Cosine Ratios OBJ: Section 3.2 NAT: M4 KEY: sine ratio | angle of elevation | Pythagorean theorem 7 Name: ________________________ Class: ___________________ Date: __________ ID: A Trigonometry Partner Assignment Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Evaluate tan 4°, to four decimal places. a. 1.6949 b. 0.9976 c. 0.6993 d. 0.0699 5. A school soccer field measures 45 m by 65 m. To get home more quickly, you decide to walk along the diagonal of the field. What is the angle of your path, with respect to the 45-m side, to the nearest degree? a. 55° b. 34° c. 2° d. 1° 2. In the triangle, BC = 12 cm and tan C = 0.583. What is the length of the hypotenuse, to the nearest tenth of a centimetre? Use the diagram to answer the following question(s). a. b. c. d. 14.2 cm 13.9 cm 7.0 cm 5.1 cm 6. Determine the length of x, to the nearest tenth of a metre. a. 5.7 m b. 6.3 m c. 7.0 m d. 8.2 m 3. Evaluate tan 45°. a. –1 b. 0 c. 1 d. undefined 7. Determine the length of y, to the nearest tenth of a metre. a. 5.7 m b. 6.3 m c. 7.0 m d. 8.2 m 4. Determine the measure of ∠C, to the nearest degree. a. b. c. d. 19° 20° 21° 22° 1 Name: ________________________ ID: A 11. Write the cosine ratio of ∠A. 8. Choose the correct formula for the cosine ratio of ∠A. length of side opposite ∠A a. cos A = length of side adjacent to ∠A length of hypotenuse b. cos A = length of side adjacent to ∠A length of side opposite ∠A c. cos A = length of hypotenuse length of side adjacent to ∠A d. cos A = length of hypotenuse a. b. 9. Determine the value of sin 28°, to four decimal places. a. 0.4695 b. 0.5317 c. 0.8829 d. 0.9709 c. d. 12. In the triangle, AC = 26 cm and AB = 10 cm. Determine the sine ratio of ∠A, to the nearest thousandth. 10. In ABC, AC = 8 cm and BC = 11 cm. Determine the sine ratio of ∠B, rounded to the nearest thousandth. a. b. c. d. AB AC BC AB AC BC BC AC a. b. c. d. 0.588 0.728 0.809 1.375 2 0.385 0.417 0.923 1.833 Name: ________________________ ID: A 13. In ABC, BC = 48 cm and sin A = 0.96. Determine the length of AC, to the nearest centimetre. a. b. c. d. Use the diagram to answer the following question(s). Kelly is flying a kite in a field. He lets out 40 m of his kite string, which makes an angle of 72° with the ground. 14 cm 24 cm 25 cm 50 cm 14. In the triangle, BC = 11 cm and cos B = 0.809. Determine the length of AC, to the nearest centimetre. a. b. c. d. 17. Determine the height of the kite above the ground, to the nearest metre. a. 12 m b. 38 m c. 42 m d. 129 m 18. Suppose the sun is shining directly above the kite. How far is the kite’s shadow from Kelly, to the nearest metre? a. 12 m b. 38 m c. 42 m d. 129 m 8 cm 9 cm 10 cm 13 cm 15. Determine the value of cos 0°. a. –1 b. 0 c. 1 d. undefined 19. An Airbus A320 aircraft is cruising at an altitude of 10 000 m. The aircraft is flying in a straight line away from Monica, who is standing on the ground. Monica observes that the angle of elevation of the aircraft changes from 70° to 33° in one minute. What is the cruising speed of the aircraft, to the nearest kilometre per hour? a. 463 km/h b. 485 km/h c. 665 km/h d. 705 km/h 16. In TUV, UV = 8 m, ∠U = 90°, and ∠T = 38°. Determine the measure of ∠V, to the nearest degree a. 42° b. 52° c. 90° d. 128° 3 Name: ________________________ ID: A 8 . 11 ABC, to the nearest square 20. In the triangle, BC = 11 cm and tan B = Determine the area of centimetre. a. b. c. d. 121 cm2 88 cm2 64 cm2 44 cm2 Completion Complete each statement. 1. The hypotenuse is the side _________________________ the right angle in a right triangle. 6. The angle between the horizontal and the line of sight looking up to an object is called the angle of _________________________ . 2. The tangent ratio of ∠A can be written as _________________________ . 7. The angle of depression is the angle between the horizontal and the line of sight looking _________________________ to an object. 3. The sine ratio of ∠A compares the length of the _________________________ side to the length of the hypotenuse. 8. The _________________________ can be used to determine the length of one side of a right triangle given the lengths of the other two sides. 4. The cosine ratio of ∠A compares the length of the _________________________ side to the length of the hypotenuse. 9. In a right triangle, the _________________________ side is the side that forms one of the arms of the angle being considered but is not the hypotenuse. 5. The angle of elevation is the angle between the horizontal and the line of sight looking _________________________ to an object. 10. The angle between the horizontal and the line of sight looking down to an object is called the angle of _________________________ . 4 Name: ________________________ ID: A Matching Match the correct side or angle of PQR to each of the following definitions, descriptions, or expressions. A term may be used more than once or not at all. a. b. c. p p q p r f. q r g. –1 h. 0 d. Pythagorean theorem i. e. q j. 1 pq 2 1 1. the side opposite ∠Q 3. r 2 = q 2 + p 2 2. the side adjacent to ∠Q 4. sin Q Match each term to the correct definition, description, or expression. A term may be used more than once or not at all. a. angle of elevation e. Pythagorean theorem b. angle of depression f. sine ratio c. cosine ratio g. tangent ratio d. hypotenuse h. trigonometry length of side opposite an acute angle length of side adjacent to an acute angle 5. in a right triangle with side lengths a, b, and c, where c is the hypotenuse, a 2 + b 2 = c 2 8. 6. the study of angles and triangles 9. the side opposite the right angle in a right triangle 7. length of side opposite an acute angle length of hypotenuse 10. 5 length of side adjacent to an acute angle length of hypotenuse Name: ________________________ ID: A Short Answer 1. Juanita is helping build a wheelchair ramp at her community centre. The ramp must reach a vertical distance of 1.5 m. If the ramp must make an angle of 6° with the ground, how long will the ramp be? Answer to the nearest tenth of a metre. 3. In right triangle ABC, the hypotenuse, AB, is 15 cm and ∠B is 25°. How long is AC, to the nearest centimetre? 2. A telephone pole is secured with a guy wire as shown in the diagram. The guy wire makes an angle of 75° with the ground and is secured to the ground 6 m from the bottom of the pole. Determine the height of the telephone pole, to the nearest tenth of a metre. 4. In XYZ, XY and XZ have equal lengths of 6 cm. YZ is 10 cm. Determine the measure of ∠X, to the nearest degree. 5. In the diagram, label the hypotenuse, the opposite side, and the adjacent side relative to ∠C. Problem 1. The percent grade of a road is the ratio of the vertical rise of the road to the horizontal distance, expressed as a percent. A road in the mountains has a vertical rise of 40 m over 1 km of horizontal distance. a) Determine the percent grade of the road, to the nearest percent. b) What is the angle of elevation of the road, to the nearest degree? 2. Myriam is participating in a water-skiing competition. She goes over a water-ski ramp that is 4.0 m long. When she leaves the ramp, she is 1.7 m above the surface of the water. a) What is the horizontal length of the ramp along the surface of the water, to the nearest tenth of a metre? b) Determine the angle of elevation of the ramp, to the nearest degree. 6 ID: A Trigonometry Partner Assignment Answer Section MULTIPLE CHOICE 1. ANS: NAT: 2. ANS: NAT: KEY: 3. ANS: NAT: 4. ANS: NAT: KEY: 5. ANS: NAT: KEY: 6. ANS: NAT: KEY: 7. ANS: NAT: KEY: 8. ANS: NAT: 9. ANS: NAT: 10. ANS: NAT: KEY: 11. ANS: NAT: KEY: 12. ANS: NAT: KEY: 13. ANS: NAT: KEY: 14. ANS: NAT: KEY: 15. ANS: NAT: 16. ANS: NAT: D PTS: 1 DIF: B OBJ: Section 3.1 M4 TOP: The Tangent Ratio KEY: tangent ratio | calculate a tangent ratio B PTS: 1 DIF: C OBJ: Section 3.1 M4 TOP: The Tangent Ratio tangent ratio | determine a distance using trigonometry | Pythagorean theorem | hypotenuse | right triangle C PTS: 1 DIF: A OBJ: Section 3.1 M4 TOP: The Tangent Ratio KEY: tangent ratio | calculate a tangent ratio C PTS: 1 DIF: B OBJ: Section 3.1 M4 TOP: The Tangent Ratio tangent ratio | determine an angle measure | right triangle A PTS: 1 DIF: C OBJ: Section 3.3 M4 TOP: Solving Right Triangles tangent ratio | determine an angle measure C PTS: 1 DIF: C OBJ: Section 3.1 M4 TOP: The Tangent Ratio tangent ratio | determine a distance using trigonometry B PTS: 1 DIF: C OBJ: Section 3.1 M4 TOP: The Tangent Ratio tangent ratio | determine a distance using trigonometry D PTS: 1 DIF: A OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios KEY: cosine ratio | define the cosine ratio A PTS: 1 DIF: A OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios KEY: sine ratio | calculate a sine ratio A PTS: 1 DIF: B OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios sine ratio | calculate a sine ratio | right triangle A PTS: 1 DIF: A OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios cosine ratio | write a cosine ratio | right triangle C PTS: 1 DIF: B OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios sine ratio | calculate a sine ratio | right triangle A PTS: 1 DIF: B OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios sine ratio | determine a distance using trigonometry | right triangle A PTS: 1 DIF: B OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios cosine ratio | determine a distance using trigonometry | right triangle C PTS: 1 DIF: A OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios KEY: cosine ratio | calculate a cosine ratio B PTS: 1 DIF: A OBJ: Section 3.2 M4 TOP: The Sine and Cosine Ratios KEY: sine ratio | determine an angle measure 1 ID: A 17. ANS: NAT: KEY: 18. ANS: NAT: KEY: 19. ANS: NAT: 20. ANS: NAT: B PTS: 1 DIF: B OBJ: Section 3.3 M4 TOP: Solving Right Triangles sine ratio | determine a distance using trigonometry | solve a right triangle A PTS: 1 DIF: C OBJ: Section 3.3 M4 TOP: Solving Right Triangles cosine ratio | determine a distance using trigonometry | solve a right triangle D PTS: 1 DIF: D OBJ: Section 3.3 M4 TOP: Solving Right Triangles KEY: sine ratio | solve a right triangle D PTS: 1 DIF: C OBJ: Section 3.3 M4 TOP: Solving Right Triangles KEY: tangent ratio | right triangle | area COMPLETION 1. ANS: opposite PTS: 1 DIF: A TOP: The Tangent Ratio 2. ANS: Example: length of side opposite ∠A tanA = length of side adjacent to ∠A OBJ: Section 3.1 NAT: M4 KEY: right triangle | hypotenuse PTS: TOP: KEY: 3. ANS: 1 DIF: B OBJ: Section 3.1 NAT: M4 The Tangent Ratio tangent ratio | write a tangent ratio | primary trigonometric ratios opposite PTS: TOP: KEY: 4. ANS: 1 DIF: A OBJ: Section 3.2 NAT: M4 The Sine and Cosine Ratios sine ratio | define the sine ratio | opposite side | primary trigonometric ratios adjacent PTS: TOP: KEY: 5. ANS: 1 DIF: A OBJ: Section 3.2 NAT: M4 The Sine and Cosine Ratios cosine ratio | define the cosine ratio | adjacent side | primary trigonometric ratios up PTS: 1 DIF: A TOP: Solving Right Triangles 6. ANS: elevation OBJ: Section 3.3 NAT: M4 KEY: angle of elevation PTS: 1 DIF: B TOP: Solving Right Triangles 7. ANS: down OBJ: Section 3.3 NAT: M4 KEY: angle of elevation PTS: 1 DIF: A TOP: Solving Right Triangles OBJ: Section 3.3 NAT: M4 KEY: angle of depression 2 ID: A 8. ANS: Pythagorean theorem PTS: 1 DIF: A TOP: Solving Right Triangles 9. ANS: adjacent OBJ: Section 3.3 NAT: M4 KEY: Pythagorean theorem PTS: 1 DIF: B TOP: The Sine and Cosine Ratios 10. ANS: depression OBJ: Section 3.2 NAT: M4 KEY: adjacent side PTS: 1 DIF: A TOP: Solving Right Triangles OBJ: Section 3.3 NAT: M4 KEY: angle of depression MATCHING 1. ANS: NAT: 2. ANS: NAT: 3. ANS: NAT: 4. ANS: NAT: 5. ANS: NAT: 6. ANS: NAT: 7. ANS: NAT: 8. ANS: NAT: KEY: 9. ANS: NAT: 10. ANS: NAT: KEY: E M4 A M4 D M4 F M4 PTS: TOP: PTS: TOP: PTS: TOP: PTS: TOP: 1 DIF: A The Tangent Ratio 1 DIF: A The Tangent Ratio 1 DIF: A The Tangent Ratio 1 DIF: B The Sine and Cosine Ratios OBJ: KEY: OBJ: KEY: OBJ: KEY: OBJ: KEY: Section 3.1 label a right triangle | opposite side Section 3.1 label a right triangle | adjacent side Section 3.1 Pythagorean theorem Section 3.2 sine ratio | primary trigonometric ratios E PTS: 1 DIF: A M4 TOP: Solving Right Triangles H PTS: 1 DIF: A M4 TOP: The Tangent Ratio F PTS: 1 DIF: B M4 TOP: The Sine and Cosine Ratios G PTS: 1 DIF: B M4 TOP: The Tangent Ratio tangent ratio | primary trigonometric ratios D PTS: 1 DIF: A M4 TOP: The Tangent Ratio C PTS: 1 DIF: B M4 TOP: The Sine and Cosine Ratios cosine ratio | primary trigonometric ratios OBJ: KEY: OBJ: KEY: OBJ: KEY: OBJ: Section 3.3 Pythagorean theorem Section 3.1 trigonometry Section 3.2 sine ratio | primary trigonometric ratios Section 3.1 3 OBJ: Section 3.1 KEY: hypotenuse OBJ: Section 3.2 ID: A SHORT ANSWER 1. ANS: Let x represent the length of the ramp, in metres. vertical height of ramp sin6° = length of ramp sin6° = x= 1.5 x 1.5 sin6° x = 14.3502. . . The wheelchair ramp must be approximately 14.4 m long. PTS: 1 DIF: B OBJ: Section 3.1 NAT: M4 TOP: The Tangent Ratio KEY: sine ratio | determine a distance using trigonometry 2. ANS: Let h represent the height of the pole, in metres. height of pole tan75° = distance of wire from base tan75° = h 6 22.3923. . . = h The pole is approximately 22.4 m tall. PTS: 1 DIF: B OBJ: Section 3.1 NAT: M4 TOP: The Tangent Ratio KEY: tangent ratio | determine a distance using trigonometry | right triangle 3. ANS: side opposite ∠B sin B = hypotenuse sin 25° = AC 15 15(sin 25°) = AC 6.3393. . . = AC AC is approximately 6 cm long. PTS: 1 DIF: B OBJ: Section 3.2 NAT: M4 TOP: The Sine and Cosine Ratios KEY: sine ratio | determine a distance using trigonometry | right triangle 4 ID: A 4. ANS: cos Z = side adjacent to ∠Z hypotenuse cos Z = ZW XZ cos Z = 5 6 cos Z = 0.8333 Z = cos −1(0.8333) Z = 33.5573. . . ∠Z measures approximately 34°. ∠Y is equal to ∠Z. 180 = ∠X + ∠Y + ∠Z 180 = ∠X + 34 + 34 180 = ∠X + 68 112 = ∠X ∠X measures approximately 112°. PTS: 1 DIF: C TOP: The Sine and Cosine Ratios 5. ANS: PTS: 1 DIF: TOP: The Tangent Ratio A OBJ: Section 3.2 NAT: M4 KEY: cosine ratio | determine an angle measure | isosceles triangle OBJ: Section 3.1 NAT: M4 KEY: primary trigonometric ratios | label a right triangle 5 ID: A PROBLEM 1. ANS: a) 1 km = 1000 m ÊÁ ˆ˜ vertical rise ˜˜ × 100 percent grade = ÁÁÁ ÁË horizontal distance ˜˜¯ ÁÊ 40 ˜ˆ˜ ˜˜ × 100 percent grade = ÁÁÁÁ ˜ Ë 1000 ¯ percent grade = 0.04 × 100 percent grade = 4 The percent grade of the road is 4%. b) Let θ represent the angle of elevation, in degrees. vertical rise tan θ = horizontal distance tan θ = 40 1000 θ = tan−1 (0.04) θ = 2.2906. . . The angle of elevation of the road is approximately 2°. PTS: 1 DIF: D TOP: Solving Right Triangles OBJ: Section 3.3 NAT: M4 KEY: tangent ratio | angle of elevation 6 ID: A 2. ANS: a) Let x represent the length of the ramp along the surface of the water, in metres. Draw a diagram of the situation. x 2 + 1.7 2 = 4.0 2 x 2 = 16 − 2.89 x 2 = 13.11 x ≈ 3.62 The horizontal length of the ramp along the surface of the water is about 3.6 m. b) sinθ = 1.7 4.0 sinθ = 0.425 θ = 25.1507. . . θ ≈ 25 The angle of elevation of the ramp is about 25°. PTS: 1 DIF: B TOP: The Sine and Cosine Ratios OBJ: Section 3.2 NAT: M4 KEY: sine ratio | angle of elevation | Pythagorean theorem 7