REVIEW, pages

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1 REVIEW, pages Determine the value of each trigonometric ratio. Use eact values where possible; otherwise write the value to the nearest thousandth. a) tan (5 ) b) cos c) sec ( ) cos º cos ( ) cos 5 d) sin 75 e) cot f) csc 8 sin 5º tan sin To the nearest degree, determine all possible values of u for which cos u.7, when u. Since cos U is positive, the terminal arm of angle U lies in Quadrant or. The reference angle is: cos (.7) º For the domain U : In Quadrant, U º In Quadrant, U º º, or approimatel 9º For the domain U : In Quadrant, U º º, or approimatel 9º In Quadrant, U º.. As a fraction of, determine the length of the arc that subtends a central angle of 5 in a circle with radius units. 5 5 Arc length: ()().. a) Convert each angle to degrees. Give the answer to the nearest degree where necessar. 5 i) ii) 7 iii) 5(8 ) (8 ) 7 a b 57 9º Chapter : Trigonometr Review Solutions DO NOT COPY. P

2 b) Convert each angle to radians. i) 5 ii) iii) 85 5a 8 b a b 85a 8 8 b In a circle with radius 5 cm, an arc of length cm subtends a central angle. What is the measure of this angle in radians, and to the nearest degree? arc length Angle measure is: radius 5. In degrees,..a 8 b 9º The angle measure is. radians or approimatel 9º.. A race car is travelling around a circular track at an average speed of km/h. The track has a diameter of 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 s? In s, the car travels: km km # So, in s, the car travels: km km arc length Angle measure is: radius In s, the segment will have rotated through an angle of 7. Determine the value of each trigonometric ratio. Use eact values where possible; otherwise write the value to the nearest thousandth. 5 a) sin b) cos c) sec a- b radian., cos a b which is undefined 5 d) tan e) csc 5 f) cot (.8) tan sin 5. tan (.8).95 P DO NOT COPY. Chapter : Trigonometr Review Solutions 5

3 8. P(, ) is a terminal point of angle u in standard position. a) Determine the eact values of all the trigonometric ratios for u. Let the distance between the origin and P be r. Use: r Substitute:, 9 r r sin U csc U cos U sec U tan U cot U b) To the nearest tenth of a radian, determine possible values of u in the domain u. The terminal arm of angle U lies in Quadrant. The reference angle is: tan a.7... b So, U.7... The angle between and that is coterminal with.7... is: Possible values of U are approimatel:. and.. 9. Use graphing technolog to graph each function below for, then list these characteristics of the graph: amplitude, period, zeros, domain, range, and the equations of the asmptotes. a) sin The amplitude is. The period is. The zeros are,,. The domain is. The range is. There are no asmptotes. b) cos The amplitude is. The period is. The zeros are,. The domain is. The range is. There are no asmptotes. c) tan There is no amplitude. The period is. The zeros are,,. The domain is,. The range is ç. The equations of the asmptotes are and. Chapter : Trigonometr Review Solutions DO NOT COPY. P

4 .5. On the same grid, sketch graphs of the functions in each pair for, then describe our strateg. a) sin and sin a - b sin sin ( ) For the graph of sin, I used the completed table of values from Lesson.. The horizontal scale is square to units, because the phase shift is. I then shifted several points units right and joined the points to get the graph of sin a. b b) cos and cos cos 5 cos For the graph of cos, I used the completed table of values from Lesson.. I multiplied ever -coordinate b.5, plotted the new points, then joined them to get the graph of cos. P DO NOT COPY. Chapter : Trigonometr Review Solutions 7

5 .. a) Graph sin a + b for. Eplain our strateg. sin ( ) sin 7 5 Sample response: I graphed sin, shifted several points units left and units up, then joined the points to get the graph of sin a. b b) List the characteristics of the graph ou drew. The amplitude is. The period is. There are no zeros. 5 The domain is. The range is.. An equation of the function graphed below has the form a cos b( c) d. Identif the values of a, b, c, and d in the equation, then write an equation for the function. f() Sample response: The equation of the centre line is, so the vertical translation is unit up and d. () 5 5 The amplitude is:, so a Choose the -coordinates of two adjacent maimum points, and. The period is: a b So, b is: To the left of the -ais, the cosine function begins its ccle at, so a possible phase shift is, and c. Substitute for a, b, c, and d in: a cos b( c) d 5 An equation is: cos a b 8 Chapter : Trigonometr Review Solutions DO NOT COPY. P

6 .7. A water wheel has diameter m and completes revolutions each minute. The ale of the wheel is 8 m above a river. a) The wheel is at rest at time t s, with point P at the lowest point on 8 m the wheel. Determine a function that models the height of P above the river, h metres, at an time t seconds. Eplain how the characteristics of the graph relate to the given information. River m P The time for revolution is 5 s. h At t, h At t 7.5, h (7.5, ) The graph begins at (, ), which is a minimum point. The first maimum point is at (7.5, ). The net minimum point is after ccle and it has coordinates (5, ). (, ) (5, ) t The position of the first maimum is known, so use a cosine function: h(t) a cos b(t c) d The constant in the equation of the centre line of the graph is the height of the ale above the river, so its equation is: h 8; and this is also the vertical translation, so d 8 The amplitude is one-half the diameter of the wheel, so a 5 The period is the time for revolution, so b 5 A possible phase shift is: c 7.5 An equation is: h(t) 5 cos (t 7.5) 8 5 b) Use technolog to graph the function. Use this graph to determine: i) the height of P after 5 s Graph: Y 5 cos (X 7.5) 8 5 Determine the Y-value when X 5. After 5 s, P is.5 m high. P DO NOT COPY. Chapter : Trigonometr Review Solutions 9

7 ii) the times, to the nearest tenth of a second, in the first 5 s of motion that P is m above the river Graph: Y 5 cos (X 7.5) 8 and Y 5 Determine the Y-coordinates of the first two points of intersection. P is m above the river after approimatel 5. s and 9.7 s. 5 Chapter : Trigonometr Review Solutions DO NOT COPY. P