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08_ch06b_pre-calculas12_wncp_solution.qxd 5/22/12 3:55 PM Page 44 Home Quit R E V I E W, p a g e s 5 6 0 – 5 6 6 6.1 1. Determine the value of each trigonometric ratio. Use exact values where possible; otherwise write the value to the nearest thousandth. a) tan (⫺45°) b) cos 600° ⴝ ⴚ1 c) sec (⫺210°) ⴝ cos 240º ⴝⴚ d) sin 765° 1 2 e) cot 21° ⴝ sin 45º 1 2 ⴝ√ 1 cos ( ⴚ 210°) 1 ⴝ cos 150° 2 ⴝ ⴚ√ 3 ⴝ f) csc 318° 1 ⴝ tan 21° ⴝ ⬟ 2.605 ⬟ ⴚ1.494 1 sin 318° 2. To the nearest degree, determine all possible values of u for which cos u ⫽ 0.76, when ⫺360° ≤ u ≤ 360°. Since cos U is positive, the terminal arm of angle U lies in Quadrant 1 or 4. The reference angle is: cosⴚ1(0.76) ⬟ 41º For the domain 0° ◊ U ◊ 360°: In Quadrant 1, U ⬟ 41º In Quadrant 4, U ⬟ 360º ⴚ 41º, or approximately 319º For the domain ⴚ360° ◊ U ◊ 0°: In Quadrant 1, U ⬟ ⴚ360º ⴙ 41º, or approximately ⴚ319º In Quadrant 4, U ⬟ ⴚ41º 6.2 3. As a fraction of , determine the length of the arc that subtends a central angle of 225° in a circle with radius 3 units. Arc length: 15 225 (2)(3) ⴝ 360 4 6.3 4. a) Convert each angle to degrees. Give the answer to the nearest degree where necessary. 5 i) 3 ⴝ 5(180°) 3 ⴝ 300° 44 10 ii) ⫺ 7 ⴝⴚ 10(180°) 7 ⬟ ⴚ257° Chapter 6: Trigonometry—Review—Solutions iii) 4 ⴝ 4a b 180° ⬟ 229º DO NOT COPY. ©P 08_ch06b_pre-calculas12_wncp_solution.qxd 5/22/12 3:55 PM Page 45 Home Quit b) Convert each angle to radians. ii) ⫺240° i) 150° b 180 b 180 ⴝ 150 a ⴝ iii) 485° b 180 ⴝ ⴚ240 a 5 6 ⴝⴚ ⴝ 485 a 4 3 ⴝ 97 36 5. In a circle with radius 5 cm, an arc of length 6 cm subtends a central angle. What is the measure of this angle in radians, and to the nearest degree? Angle measure is: arc length radius ⴝ 6 5 ⴝ 1.2 180° In degrees, 1.2 ⴝ 1.2 a b ⬟ 69º The angle measure is 1.2 radians or approximately 69º. 6. A race car is travelling around a circular track at an average speed of 120 km/h. The track has a diameter of 1 km. Visualize a line segment joining the race car to the centre of the track. Through what angle, in radians, will the segment have rotated in 10 s? 1 120 km ⴝ km 60 # 60 30 1 10 So, in 10 s, the car travels: km ⴝ km 30 3 1 arc length 3 Angle measure is: ⴝ 1 radius 1 ⴝ 3 In 1 s, the car travels: In 10 s, the segment will have rotated through an angle of 1 radian. 3 7. Determine the value of each trigonometric ratio. Use exact values where possible; otherwise write the value to the nearest thousandth. 5 a) sin 3 b) cos 6 √ ⴝ √ 3 2 ⴝⴚ 3 2 c) sec a- 2 b ⴝ 1 cos aⴚ b 2 , which is undefined 15 d) tan 4 ⴝ tan 3 4 ⴝ ⴚ1 ©P DO NOT COPY. e) csc 5 ⴝ 1 sin 5 ⬟ ⴚ1.043 f) cot (⫺22.8) ⴝ 1 tan ( ⴚ 22.8) ⬟ ⴚ0.954 Chapter 6: Trigonometry—Review—Solutions 45 08_ch06b_pre-calculas12_wncp_solution.qxd 5/22/12 3:55 PM Page 46 Home Quit 8. P(3, ⫺1) is a terminal point of angle u in standard position. a) Determine the exact values of all the trigonometric ratios for u. Let the distance between the origin and P be r. Substitute: x ⴝ 3, y ⴝ ⴚ1 Use: x2 ⴙ y2 ⴝ r2 9 ⴙ 1 ⴝ r2 √ r ⴝ 10 √ 3 1 sin U ⴝ ⴚ √ csc U ⴝ ⴚ 10 cos U ⴝ √ 10 √ 10 sec U ⴝ 3 10 tan U ⴝ ⴚ 1 3 cot U ⴝ ⴚ3 b) To the nearest tenth of a radian, determine possible values of u in the domain ⫺2 ≤ u ≤ 2. The terminal arm of angle U lies in Quadrant 4. The reference angle is: tanⴚ1 a b ⴝ 0.3217. . . 1 3 So, U ⴝ ⴚ0.3217. . . The angle between 0 and 2 that is coterminal with ⴚ0.3217. . . is: 2 ⴚ 0.3217. . . ⴝ 5.9614. . . Possible values of U are approximately: 6.0 and ⴚ0.3 6.4 9. Use graphing technology to graph each function below for ⫺2 ≤ x ≤ 2, then list these characteristics of the graph: amplitude, period, zeros, domain, range, and the equations of the asymptotes. a) y ⫽ sin x The amplitude is 1. The period is 2. The zeros are 0, — , — 2. The domain is ⴚ2 ◊ x ◊ 2. The range is ⴚ1 ◊ y ◊ 1. There are no asymptotes. b) y ⫽ cos x 2 The amplitude is 1. The period is 2. The zeros are — , — The domain is ⴚ2 asymptotes. 3 . 2 ◊ x ◊ 2. The range is ⴚ1 ◊ y ◊ 1. There are no c) y ⫽ tan x There is no amplitude. The period is . The zeros are 0, — , — 2. 3 . The range is y ç ⺢. The equations of 2 3 the asymptotes are x ⴝ — and x ⴝ — . 2 2 2 The domain is x ⴝ — , — 46 Chapter 6: Trigonometry—Review—Solutions DO NOT COPY. ©P 08_ch06b_pre-calculas12_wncp_solution.qxd 5/22/12 3:55 PM Home Page 47 Quit 6.5 10. On the same grid, sketch graphs of the functions in each pair for 0 ≤ x ≤ 2, then describe your strategy. a) y ⫽ sin x and y ⫽ sin ax - 4 b 1 y y sin x 2 0 1 x 3 2 y sin x 4 ( ) For the graph of y ⴝ sin x, I used the completed table of values from Lesson 6.4. The horizontal scale is 1 square to I then shifted several points units, because the phase shift is . 4 4 units right and joined the points to get 4 4 the graph of y ⴝ sin ax ⴚ b . 3 b) y ⫽ cos x and y ⫽ 2 cos x 3 y cos x 2 y 1 0 1 y cos x x 3 5 3 For the graph of y ⴝ cos x, I used the completed table of values from Lesson 6.4. I multiplied every y-coordinate by 1.5, plotted the new points, then joined them to get the graph of y ⴝ ©P DO NOT COPY. 3 cos x. 2 Chapter 6: Trigonometry—Review—Solutions 47 08_ch06b_pre-calculas12_wncp_solution.qxd 5/22/12 3:55 PM Page 48 Home Quit 6.6 1 11. a) Graph y ⫽ 2 sin 3 ax + 6 b ⫹ 2 for ⫺2 ≤ x ≤ 2. Explain your strategy. 3 y y 1 2 sin 3 x 6 2 ( ) 2 y 1 sin 3x 2 1 x 0 11 6 2 7 6 6 1 Sample response: I graphed y ⴝ 5 6 3 2 1 sin 3x, shifted several points units 2 6 left and 2 units up, then joined the points to get the graph of yⴝ 1 sin 3 ax ⴙ b ⴙ 2. 2 6 b) List the characteristics of the graph you drew. 1 2 2 . There are no zeros. 3 3 5 2. The range is ◊ y ◊ . 2 2 The amplitude is . The period is The domain is ⴚ2 ◊x◊ 12. An equation of the function graphed below has the form y ⫽ a cos b(x ⫺ c) ⫹ d. Identify the values of a, b, c, and d in the equation, then write an equation for the function. y 2 y f(x) x 0 2 3 4 2 1 2 Sample response: The equation of the centre line is y ⴝ , so the 1 unit up and d ⴝ 2 3 ⴚ ( ⴚ 2) 5 The amplitude is: ⴝ , so a ⴝ 2 2 vertical translation is 1 . 2 5 2 Choose the x-coordinates of two adjacent maximum points, 11 11 and . The period is: ⴚ aⴚ b ⴝ 4 3 3 3 3 2 1 So, b is: ⴝ 4 2 ⴚ To the left of the y-axis, the cosine function begins its cycle at 3 3 3 x ⴝ ⴚ , so a possible phase shift is ⴚ , and c ⴝ ⴚ . Substitute for a, b, c, and d in: y ⴝ a cos b(x ⴚ c) ⴙ d An equation is: y ⴝ 48 1 5 1 cos ax ⴙ b ⴙ 2 2 3 2 Chapter 6: Trigonometry—Review—Solutions DO NOT COPY. ©P 08_ch06b_pre-calculas12_wncp_solution.qxd 5/22/12 3:55 PM Page 49 Home Quit 6.7 13. A water wheel has diameter 10 m and completes 4 revolutions each minute. The axle of the wheel is 8 m above a river. 10 m a) The wheel is at rest at time t ⫽ 0 s, with point P at the lowest point on the wheel. Determine a function that 8 m models the height of P above the river, h metres, at any time t seconds. Explain how the characteristics of the graph relate to the given information. P River The time for 1 revolution is 15 s. h At t ⴝ 0, h ⴝ 3 At t ⴝ 7.5, h ⴝ 13 (7.5, 13) The graph begins at (0, 3), which is a minimum point. The first maximum point is at (7.5, 13). (15, 3) The next minimum point is after 1 cycle (0, 3) and it has coordinates (15, 3). t The position of the first maximum is 0 known, so use a cosine function: h(t) ⴝ a cos b(t ⴚ c) ⴙ d The constant in the equation of the centre line of the graph is the height of the axle above the river, so its equation is: h ⴝ 8; and this is also the vertical translation, so d ⴝ 8 The amplitude is one-half the diameter of the wheel, so a ⴝ 5 2 The period is the time for 1 revolution, so b ⴝ 15 A possible phase shift is: c ⴝ 7.5 2 An equation is: h(t) ⴝ 5 cos (t ⴚ 7.5) ⴙ 8 15 b) Use technology to graph the function. Use this graph to determine: i) the height of P after 35 s Graph: Y ⴝ 5 cos 2 (X ⴚ 7.5) ⴙ 8 15 Determine the Y-value when X ⴝ 35. After 35 s, P is 10.5 m high. ©P DO NOT COPY. Chapter 6: Trigonometry—Review—Solutions 49 08_ch06b_pre-calculas12_wncp_solution.qxd 5/22/12 3:55 PM Home Page 50 Quit ii) the times, to the nearest tenth of a second, in the first 15 s of motion that P is 11 m above the river Graph: Y ⴝ 5 cos 2 (X ⴚ 7.5) ⴙ 8 and Y ⴝ 11 15 Determine the Y-coordinates of the first two points of intersection. P is 11 m above the river after approximately 5.3 s and 9.7 s. 50 Chapter 6: Trigonometry—Review—Solutions DO NOT COPY. ©P