Unit two review (trig)

Transcription

1 Class: Date: Unit two review (trig) Multiple Choice Identify the choice that best completes the statement or answers the question. 1. What is the reference angle for 15 in standard position? A 255 C 345 B 30 D What is the exact sine of A? A 1/ 3 C 2/ 3 B 1/3 D 1/2 3. Which set of angles has the same terminal arm as 40? A 80, 120, 160 C 200, 380, 560 B 130, 220, 310 D 400, 760, The point (40, 9) is on the terminal arm of A. Which is the set of exact primary trigonometric ratios for the angle? A sin A = 41 41, cos A = 9 40, tan A = 9 40 B sin A = 40 41, cos A = 9, tan A = C sin A = 40 41, cos A = 9 41, tan A = 9 40 D sin A = 9 40, cos A = 41 41, tan A = Marco is 450 m due east of the centre of the park. His friend Ray is 450 m due south of the centre of the park. Which is the correct expression for the exact distance between the two boys? A m C m 225 B 2 m D m 1

2 6. Solve to the nearest tenth of a unit for the unknown side in the ratio a sin30 = 12 sin115. A 24 C 6.6 B 21.8 D Determine the length of x, to the nearest tenth of a centimetre. A 26.6 C 11.2 B 36.5 D Determine, to the nearest tenth of a centimetre, the two possible lengths of a. A 72.8 cm and 26.3 cm C 72.8 cm and 55.8 cm B 34.3 cm and 26.3 cm D 55.8 cm and 34.3 cm 2

3 9. Determine, to the nearest tenth of a degree, the two possible measures of C. A 9 and 81 C and 0.4 B and 171 D 9 and 171 3

4 10. Which of the following triangles cannot be solved using the sine law? Diagrams not drawn to scale. A C B D 4

5 11. If B = 58.8, c = 10.3 cm, and b = 10.5 cm, and ABC is acute, what is the measure of C, to the nearest tenth of a degree? A 57 C 30.5 B D Which strategy would be best to solve for x in the triangle shown? A cosine law C sine law B primary trigonometric ratios D none of the above 5

6 13. Determine the measure of x, to the nearest tenth of a degree. A 25.6 C B 18.1 D What is the length of x, to the nearest tenth of a metre? A 27.7 m C 26.1 m B 21.8 m D 37.6 m 6

7 15. If Q = 31, r = 20 cm, and p = 23 cm, what is the length of q, to the nearest centimetre? A 21 cm C 12 cm B 30 cm D 11 cm 16. Solve the following triangle, rounding side lengths to the nearest tenth of a unit and angle measures to the nearest degree. Diagram not drawn to scale. A = 152, b = 19, a = 23.5 A B = 22, C = 6, c = 5.0 B B = 158, C = 84, c = 5.0 C B = 68, C = 174, c = 28.7 D B = 35, C = 7, c =

8 17. While flying, a helicopter pilot spots a water tower that is 7.4 km to the north. At the same time, he sees a monument that is 8.5 km to the south. The tower and the monument are separated by a distance of 11.4 km along the flat ground. What is the angle made by the water tower, helicopter, and monument? A 91 C 40 B 11 D 48 Completion Complete each statement. 1. The expression cos 30 is equivalent to sin. 2. An angle between 0 and 360 that has the same sine value as sin 133 is. 3. The tangent ratio is positive in the first and quadrants. 4. The law is used to solve a triangle when two sides and their contained angle are given. 5. The sine law for acute PQR states that Q. Short Answer 1. a) For the given trigonometric ratio, determine two other angles that give the same value. i) sin 45 ii) tan 300 iii) cos 120 b) Explain how you determined the angles in part a). 8

9 Problem 1. Consider A such that cos A = a) In which quadrant(s) is this angle? Explain. b) If the sine of the angle is negative, in which quadrant is the angle? Explain. c) Sketch a diagram to represent the angle in standard position, given that the condition in part b) is true. d) Find the coordinates of a point on the terminal arm of the angle. e) Write exact expressions for the other two primary trigonometric ratios for the angle. 2. Two wires are connected to a tower at the same point on the tower. Wire 1 makes an angle of 45 with the ground and wire 2 makes an angle of 60 with the ground. a) Represent this situation with a diagram. b) Which wire is longer? Explain. c) If the point where the two wires connect to the tower is 35 m above the ground, determine exact expressions for the lengths of the two wires. d) Determine the length of each wire, to the nearest tenth of a metre. e) How do your answers to parts b) and d) compare? 9

10 Unit two review (trig) Answer Section MULTIPLE CHOICE 1. ANS: D PTS: 1 DIF: Easy OBJ: Section 2.1 NAT: T 1 TOP: Angles in Standard Position KEY: reference angle < ANS: D PTS: 1 DIF: Average OBJ: Section 2.1 NAT: T 1 TOP: Angles in Standard Position KEY: special angles sine 3. ANS: D PTS: 1 DIF: Easy OBJ: Section 2.1 NAT: T 1 TOP: Angles in Standard Position KEY: co-terminal angles 4. ANS: D PTS: 1 DIF: Average OBJ: Section 2.2 NAT: T 1 TOP: Trigonometric Ratios of Any Angle KEY: point on terminal arm cosine sine tangent 5. ANS: D PTS: 1 DIF: Easy OBJ: Section 2.2 NAT: T 1 TOP: Trigonometric Ratios of Any Angle KEY: tangent 6. ANS: C PTS: 1 DIF: Easy OBJ: Section 2.3 NAT: T 3 TOP: The Sine Law KEY: sine law side length 7. ANS: B PTS: 1 DIF: Easy OBJ: Section 2.3 NAT: T 3 TOP: The Sine Law KEY: sine law side length 8. ANS: B PTS: 1 DIF: Difficult OBJ: Section 2.3 NAT: T 3 TOP: The Sine Law KEY: sine law side length ambiguous case 9. ANS: C PTS: 1 DIF: Difficult OBJ: Section 2.3 NAT: T 3 TOP: The Sine Law KEY: sine law angle measure ambiguous case 10. ANS: A PTS: 1 DIF: Average OBJ: Section 2.3 NAT: T 3 TOP: The Sine Law KEY: sine law < ANS: A PTS: 1 DIF: Average OBJ: Section 2.3 NAT: T 3 TOP: The Sine Law KEY: sine law angle measure 12. ANS: C PTS: 1 DIF: Easy OBJ: Section 2.3 NAT: T 3 TOP: The Sine Law KEY: sine law solution method 13. ANS: B PTS: 1 DIF: Average OBJ: Section 2.4 NAT: T 3 TOP: The Cosine Law KEY: cosine law angle measure 14. ANS: C PTS: 1 DIF: Average OBJ: Section 2.4 NAT: T 3 TOP: The Cosine Law KEY: cosine law side length 15. ANS: C PTS: 1 DIF: Average OBJ: Section 2.4 NAT: T 3 TOP: The Cosine Law KEY: cosine law side length 16. ANS: A PTS: 1 DIF: Difficult OBJ: Section 2.3 Section 2.4 NAT: T 3 TOP: The Sine Law The Cosine Law KEY: cosine law sine law solve a triangle 17. ANS: A PTS: 1 DIF: Average OBJ: Section 2.4 NAT: T 3 TOP: The Cosine Law KEY: cosine law angle measure 1

11 COMPLETION 1. ANS: 60 PTS: 1 DIF: Easy OBJ: Section 2.1 NAT: T 1 TOP: Angles in Standard Position KEY: cosine sine special angles 2. ANS: 47 PTS: 1 DIF: Average OBJ: Section 2.2 NAT: T 1 TOP: Trigonometric Ratios of Any Angle KEY: sine reference angle 3. ANS: third or 3rd PTS: 1 DIF: Easy OBJ: Section 2.2 NAT: T 2 TOP: Trigonometric Ratios of Any Angle KEY: ratio quadrant 4. ANS: cosine PTS: 1 DIF: Average OBJ: Section 2.3 Section 2.4 NAT: T 3 TOP: The Sine Law The Cosine Law KEY: cosine law sine law Ê q sinr ˆ Ê q sinp ˆ 5. ANS: sin 1 or sin 1 r Ë Á p Ë Á PTS: 1 DIF: Average OBJ: Section 2.3 NAT: T 3 TOP: The Sine Law KEY: sine law SHORT ANSWER 1. ANS: a) Answers may vary. Sample answers: i) 315 and 405 ii) 60 and 120 iii) 240 and 240 b) Sketch the given angle on a Cartesian plane, and identify its reference angle. Then, determine the other quadrant where the trigonometric ratio has the same sign as the given ratio and reflect the reference angle into that quadrant. Any angle co-terminal to the two angles in the diagram will have the same trigonometric ratio as that given. PTS: 1 DIF: Easy OBJ: Section 2.1 Section 2.2 NAT: T 1 T 2 TOP: Angles in Standard Position Trigonometric Ratios of Any Angle KEY: primary trigonometric ratios reference angle 2

12 PROBLEM 1. ANS: a) Since the cosine ratio is positive, the angle is in the first or the fourth quadrant. b) If the sine ratio is negative, the angle is located in the fourth quadrant. c) d) Use the Pythagorean theorem. r 2 = x 2 + y = y 2 y 2 = = 25 y = ±5 Since the point is in the fourth quadrant, y = 5. Therefore, a point on the terminal arm is (12, 5). e) sina = 5 13,tanA = 5 12 PTS: 1 DIF: Average OBJ: Section 2.1 Section 2.2 NAT: T 1 T 2 TOP: Angles in Standard Position Trigonometric Ratios of Any Angle KEY: primary trigonometric ratios standard angle 3

13 2. ANS: a) b) Both wires are connected to the tower at the same height, which is the opposite side to the given angles. Each wire length represents the hypotenuse of its respective triangle. The longer hypotenuse is the wire that forms the smaller angle, as it will need to be longer to reach the tower. c) Let x represent the length of wire 1 and y represent the length of wire 2. Wire 1: sin45 = 35 x x = = 35 sin45 Ê Ë Á ˆ Wire 2: sin60 = 35 y y = = 35 sin60 Ë Á = 35 2 = 70 3 d) The length of wire 1 is 49.5 m, and the length of wire 2 is 40.4 m. e) The values calculated in part d) support the answers in part b). PTS: 1 DIF: Average OBJ: Section 2.1 Section 2.2 NAT: T 1 T 2 TOP: Angles in Standard Position Trigonometric Ratios of Any Angle KEY: special angles sine Ê ˆ 4

14 Unit two review (trig) [Answer Strip] C 6. C 9. A 10. A 11. D 1. B 7. D 2. C 12. B 8. D 3. D 4. D 5.

15 Unit two review (trig) [Answer Strip] B 13. C 15. A 17. C 14. A 16.