UNIT 6 SINUSOIDAL FUNCTIONS Date Lesson Text TOPIC Homework Ma 0 6. (6) 6. Periodic Functions Hula Hoop Sheet WS 6. Ma 4 6. (6) 6. Graphing Sinusoidal Functions Complete lesson shell WS 6. Ma 5 6. (6) 6. Graphing Sinusoidal Functions II WS 6. Ma 6 6.4 (64) Writing Equations of Sinusoidal Functions WS 6.4 Ma 7 6.5 (65) Trig Equations QUIZ (6. 6.) WS 6.5 Ma 8 6.6 (66) Applications of Trig Functions WS 6.6 Ma 6.7 (67) Applications of Trig Functions II Pg. 7 # 0 Pg. 98 #,, 5, 6 Ma 6.8 (68) Reciprocal Trig Ratios WS 6.8 # 6, 8, 9,, 6, 8, 9 Ma 4 6.9 (57) 5.5 Trig Identities QUIZ (6.4 6.7) Pg. 0 # - 5, 7abc, 8, Ma 5 6.0 (58) 5.5 Trig Identities WS 6.0 Ma 8 6. (69) Review for Unit 6 Test WS 6. Ma 0 6. (70) UNIT 6 TEST
MCRU Lesson 6. Introduction to Periodic Functions Fredd the Frog is riding on the circumference of a mill wheel on a mini-putt course as it rotates counter-clockwise. He would like to know the relationship between the angle of rotation and his height above / below the surface of the water. The wheel has a radius of metre and exactl half of the wheel is under water. You can use the wheel at right as necessar to help enter the values in the table below. Angle (degrees) Height (metres) Angle (degrees) Height (metres) 0 5 0 45 60 75 90 05 0 5 50 65 80 95 0 5 40 55 70 85 00 5 0 45 60 On the grid provided, draw the graph relating the angle with the height. height.5 0.5 60 60 0 80 40 00 60 angle 0.5.5 The above function is, because after one rotation. For this function, the part from to is called a. The length of one is called the. For the above function: Period: Amplitude: Axis of the Curve:
On the grids below sketch the function if the radius of the wheel is changed to: height a) metres. b) 0.5 metres. height 60 angle 60 angle Also state the: Amplitude: Period: Axis of the Curve: Amplitude: Period: Axis of the Curve: If it takes 4 seconds for this wheel ( metre radius) to make a revolution, use the grid below to sketch the graph of this function for two complete ccles. Assume that Freddie jumps on at water level. height 4 48 time
On the same grid below, plot one ccle if it takes a) seconds and b) 48 seconds to make a revolution. height 4 48 angle For (a), state the amplitude, period, and axis of the curve. For (b), state the amplitude, period, and axis of the curve. If Fredd (after a spectacular leap) begins at point B instead of point A, sketch the graph representing his path for one 4 second ccle. B height Amplitude: Period: C A 4 time Axis of Curve: D Phase Shift: How about starting at C? How about Frannie the fish, who started at D? height height 4 time 4 time Amplitude: Period: Axis of Curve: Phase Shift: Amplitude: Period: Axis of Curve: Phase Shift:
MCRU Working With Periodic Functions. For the periodic function, f, pictured below, a) sketch the function for two more ccles. 4 9 8 7 6 5 4 4 5 6 7 8 9 0 4 x 4 b) State the Period. Amplitude. Axis of the Curve Value of f() f(7) f(-7) f(64). For the periodic function, f, pictured below, state the 4 5 4 4 5 6 7 8 9 0 4 5 6 7 8 9 x 4 Period. Amplitude. Axis of the Curve Value of f() f(0) f(5) f(94). For the periodic function, f, pictured at right, state the 5 Period Amplitude Axis of the Curve 4 5 4 4 5 6 7 8 9 0 x 4 5 Value of f(0) f(7) f(50) f(9) WS 6. plus HW: Pg. 5 #,, 4, 6, 7, 8, 9
MCRU Lesson 6. Graphing Sinusoidal Functions I In Lesson, we graphed the relationship of Freddie the Frog s height on a wheel versus the angle at which he was positioned. We will transfer this data to the table of values for the relation sin. Following the table is the graph of the relation sin. (degrees) 0 5 0 45 60 75 90 05 0 5 50 65 0.6.5.7.87.97.97.87.7.5.6 (degrees) 80 95 0 5 40 55 70 85 00 5 0 45 60 0 -.6 -.5 -.7 -.87 -.97 - -.97 -.87 -.7 -.5 -.6 0 90 80 70 60 450 540 60 70 Complete the following table of values and graph the relation cos. (degrees) 0 5 0 45 60 75 90 05 0 5 50 65 80 (degrees) 95 0 5 40 55 70 85 00 5 0 45 60 80 90 90 80 70 60 450 540 60 70 x = sinx
Sinusoidal Functions: Functions transformed from the graphs of sin and cos. Transformation #: a sin, a cos On the same axes sketch one ccle of sin, sin, sin. 80 90 90 80 70 60 450 540 60 70 On the same axes sketch one ccle of cos, cos, cos. 80 90 90 80 70 60 450 540 60 70 On the same axes sketch one ccle of sin, sin 80 90 90 80 70 60 450 540 60 70
On the same axes sketch one ccle of cos, cos. 80 90 90 80 70 60 450 540 60 70 Summar (for the transformations a sin and a cos ): If a, there is a. If 0 a, there is a. If a 0, there is also a. a, the absolute value of a is the. Transformation #: sin vs, cos vs On the same axes sketch one ccle of sin, sin. 80 90 90 80 70 60 450 540 60 70 On the same axes sketch one ccle of cos, cos. 80 90 90 80 70 60 450 540 60 70
Summar (for the transformations sin vs and cos vs): If vs 0, there is a. If vs 0, there is a. Transformation #: sin ps, cos ps On the same axes sketch one ccle of sin, sin 90. 80 90 90 80 70 60 450 540 60 70 On the same axes sketch one ccle of cos, cos 45. 80 90 90 80 70 60 450 540 60 70 Summar (for the transformations sin ps and ps cos ): If ps 0, there is a. If ps 0, there is a. ps represents the of the function.
Complete the following chart: Function Amplitude (a) Phase Shift (ps) Vertical Shift (vs) sin 0.5 cos 60 cos sin 0 sin 45 Now graph two ccles of each of the above functions: sin 0.5 80 90 90 80 70 60 450 540 60 70 cos 60 80 90 90 80 70 60 450 540 60 70 cos 80 90 90 80 70 60 450 540 60 70
sin 0 80 90 90 80 70 60 450 540 60 70 sin 45 80 90 90 80 70 60 450 540 60 70 HOMEWORK 6. : HW: WS 6. Complete the following chart: Function Amplitude (a) Phase Shift (ps) Vertical Shift (vs) cos sin 60 sin cos 0. 5 80 cos sin 5 On WS 6., graph at least one ccle of each of the above functions:
MCRU Lesson 6. Graphing Sinusoidal Functions II Transformation #4: sink, cosk On the same axes sketch one ccle of sin, sin, sin HW: WS 6. 80 90 90 80 70 60 450 540 60 70 On the same axes sketch one ccle of cos, cos. 80 90 90 80 70 60 450 540 60 70 Summar (for the transformations sink and cosk ): If k, there is a. If 0 k, there is a. Function Period Function Period sin cos sin cos sin cos 5 sin cos 4 The period of sin( k ) and cos( k ) is:
For each of the following, complete the table and sketch for ccles: Function sin.5 Amplitude (a) Period Phase Shift (ps) Vertical Shift (vs) 80 90 90 80 70 60 450 540 60 70 Function cos 80 Amplitude (a) Period Phase Shift (ps) Vertical Shift (vs) 80 90 90 80 70 60 450 540 60 70 sin Function 0 Amplitude (a) Period Phase Shift (ps) Vertical Shift (vs) 80 90 90 80 70 60 450 540 60 70
MCRU Lesson 6.4 Finding the Equation of a Sinusoidal Function The general equation of the sine function is a sin kx p. s. v. s. phase shift is p.s. and the period is 60. Note that since k. Find the equation of the following sine functions: a) amplitude =, period = 60, phase shift = 0 b) amplitude =, period = 80, phase shift = 45, where the amplitude is a, the 60 period, therefore k. k c) amplitude =, period = 540, phase shift = 0 d) amplitude =, period = 90, phase shift = 80. For each graph shown, determine the amplitude, period, and phase shift; then state the defining equation of the sine function. a) amplitude: period: phase shift: 80 80 60 equation: b) amplitude: period: phase shift: 80 80 60 equation: c) amplitude: period: phase shift:.5 0.5 80 80 60 equation: 0.5.5
d) amplitude: period: 80 80 60 phase shift: equation:. For each of the graphs shown in #, determine the amplitude, period, and phase shift; then state the defining equation of the cosine function. a) amplitude: period: phase shift: equation: 80 80 60 b) amplitude: period: phase shift: 80 80 60 equation:.5 c) amplitude: period: 0.5 phase shift: 80 0.5 80 60 equation:.5 d) amplitude: period: phase shift: 80 80 60 equation:
4. For each of the graphs shown in #4, determine the amplitude, period, and phase shift; then state the defining equation of the either the sine or the cosine function, (whichever is more appropriate). a) amplitude: period: phase shift: axis of the curve: equation: 80 90 90 80 70 60 x b) amplitude: period: phase shift: axis of the curve: 80 90 90 80 70 60 x equation: c) amplitude: period: phase shift: axis of the curve: equation: 80 90 90 80 70 60 x d) amplitude: period: phase shift: axis of the curve: 80 90 90 80 70 60 x HW: WS 6.4 equation:
MCRU Lesson 6.5 Solving Trigonometric Equations. Solve each equation for 0 60. a) sin tan b) c) cos 5. Solve each equation for 0 60. sin sin a) 0 b) cos cos 0. Solve each equation for 0 60. a) cos 0 b) sin 4. Solve each equation for 0 60. b) sin 40. 8988 a) sin 0. 8988 c) cos 0. 78 HW: WS 6.5
MCRU Lesson 6.6 Applications of Sinusoidal Functions. A weight is supported b a spring. The weight rests 50 cm above a tabletop. The weight is pulled down 0 cm and released at time t = 0. This creates a periodic upand-down motion. It takes.6s for the weight to return to the low position each time. Sketch a graph of the height of the weight above the tabletop against time. 00 80 60 h Assuming a sinusoidal relationship between height and time, write an equation relating height, h, and time, t. 40 0 4 t a) What will be the height of the weight after i).8 seconds? ii).5 seconds b) At what time will the height of the weight be 60 cm?
. The depth of water in a ba varies according to the tides. A pole is placed in the water to measure the water s depth. At high tide (midnight) the water at the pole is m deep. At low tide the water at the pole is m deep. Assume that the tides run in a hour sinusoidal ccle. a) How deep is the water at the pole at :00 a.m? b) How deep is the water at the pole at 4:45 p.m? c) At what times during the da will the depth of the water be 0 m? t 4 0 9 8 7 6 5 4 4 5 6 7 8 9 0 4 5 6 7 8 9 0 4 h HW: WS 6.6
MCR U Lesson 6.7 Applications of Sinusoidal Functions II Ex... Proportion of Moon Visible 0.9 0.8 0.7 0.6 0.5 0.4 0. 0. 0. 4 6 8 0 4 6 8 0 4 6 8 0 4 6 8 40 4 44 46 48 50 5 54 56 x Da of Year Pg. 7 # 0 and Pg. 98 #,, 5, 6
MCR U Lesson/WS 6.8 Reciprocal Trigonometric Ratios Ex.. The point Q( 9, ) lies on the terminal arm of. Calculate: a) cosecant b) secant c) cotangent Ex.. is a third quadrant angle, and tan =. Find the reciprocal trig values. 4 Ex.. is a first quadrant angle. If cos =, find sec. 5 Ex. 4. Draw each angle in standard position. Calculate the other reciprocal trig ratios of each angle. a) sec =, in rd quadrant b) csc = 7 8, in 4th quadrant Ex. 5. is a nd quadrant angle and csc = 7. Find a value for sin + cos. 5
Ex. 6 Find, to the nearest degree, for each of the following. 0 90 a) cot 0. 89 b) csc. 478 Ex. 7 Find, to the nearest degree, for each of the following. 0 60 a) sec. 857 b) cot 0. 99 Ex. 8 Find the value of each of the following, correct to decimal places. a) csc 4 b) sec c) cot 7 ---------------------------------------------------------------------------------------------------------------------------------- WS 6.8 6. For each angle, a point on the terminal arm is shown. Calculate the reciprocal trig ratios. HW: WS 6.8 # 6, 8, 9,,, 6, 7, 8 a) b) (, 5) (, ) (, ) 7. is an angle in the nd quadrant and csc = 7 5. a) Draw in standard position and label a point on the terminal arm. b) Find the value of sec and cot. 8. Given that sec = 5 7, a) In which quadrants can the terminal arm be placed? b) Draw a diagram for each case in a). c) Calculate the trig ratios of csc. 9. a) Given that cot = 5. Find values of sin. b) For sec = 5 7, find tan.
0. Draw a sketch of 5 in standard position and calculate the reciprocal trig values of 5.. Calculate the exact value of each of the following. a) sec ( 0 ) b) cot 0 c) sec ( 45 ) d) csc 40 e) csc ( 495 ) f) csc 5 sin ( 0 ) cot 5. For each of the following, 0 60. Find the possible values of. a) cot = b) csc = c) sec =. Find the value of each of the following correct to 4 decimal places. a) csc 45 b) sec 47 c) cot 45 d) csc 54 e) cot 54 f) csc 5 g) sec ( 57 ) h) cot ( 7 ) 4. As increases from 45 to 54 does the value of each of the following increase or decrease? Explain. a) csc b) sec c) cot 5. Find, to the nearest degree, for each of the following. 0 90 a) cot = 0.89 b) sec =.64 c) sec =.478 d) csc =.478 e) cot = 0.99 6. For each, the quadrant is given. Find the value of, 0 60. a) csc =.547, III b) sec =.744, I 7. Find two possible values for for each of the following 0 60. a) sec =.0785 b) cot = 0.09 c) csc =.89 8. Without referring to the tables of trigonometric values or using our calculator, which of the following are false, for 0 90? Give reasons for our answer. a) cos =.5 b) tan = 0.5 c) cot =.5 d) sec = 0.5 e) sin = -.5 f) csc = 5.5 Answers. a) 0 b) sec 7 b) 0 c) 9 9 5, cot 5 8 5. 6 7. a) csc 5 b) sec 5 4 c) cot 4 6. a) csc 4. sec 4, cot 5 5 c) csc. sec, cot 7. a) ( 8, 5) b) sec 7 8, cot 8 5. sec 5 4. a) csc, cot 5 5 b) csc. sec, cot 8. a) II, III b) 5 4 9. a) sin 5 b) tan 4 0. csc. sec, cot. a) b) c) d) 6 e) f). a) 45, 5 7 b) 5, 5 c) 0, 40. a).44 b).466 c).0000 d).6 e) 0.765 f).868 g).86 h) 7.54 4. a) D b) I c) D 5. a) 50 b) 5 c) 47 d) 48 e) 47 6. a) 40 b) 55 7. a), 8 b) 0, 8 c) 58, 8. ade
MCRU Lesson 6.9 Trigonometric Identities The Pthagorean Identities: sin cos The Reciprocal Identities: sin cos csc sin sin csc cos sin sec cos cos sec cot cot tan tan sin The Quotient Identities: tan cos cos cot sin Strategies that ma help prove trigonometric identities:. Move from the complex to the simple. Sometimes ou ma simplif both sides at the same time to get the same expression. cos x ( sinx)( sinx). Ex : Prove Start with the R.S. and make it equal the L.S. R.S. ( sinx)( sinx) sinx sinx sin x sin x cos x L.S.. Look for squares and use the Pthagorean identities. Ex : Prove that R.S. ( cos ) (sin ) sin 6 L.S. sin 6 ( cos ). Express functions in terms of sin x and cos x. Ex : Prove cos θ tan θ sin θ L. S. cos θ tan θ cos θ sin θ R. S. sin θ cos θ 4. Algebraic manipulations of distribution (expanding), factoring, and creating common denominators ma allow cancellations in order to get to simpler expressions. Ex 4 : Prove that (sin cos) (sin cos). L. S. sin sin (sin () β sin β cos β cos β cos β cos β β) β sin β sin β cos β cos β 5. The most important strategies of all are patience, perseverance and ingenuit. If ou get stuck, start again with a different outlook and plan. Good Luck!
. Simplif each of the following, writing in terms of sin and/or cos. a) cos b) tan c) tan sin d) cos tan e) sin cos f) sin sin. Prove each of the following identities. a) sin sin tan
sin cos b) sin cos sin cos sin cos c) sin cos tan sin d) cot csc Pg. 0 # - 5, 7abc, 8,