Chapter 14 Trig Graphs and Reciprocal Functions Algebra II Common Core LESSON 1: BASIC GRAPHS OF SINE AND COSINE LESSON : VERTICAL SHIFTING OF SINUSOIDAL GRAPHS LESSON 3 : THE FREQUENCY AND PERIOD OF A SINUSOIDAL GRAPH LESSON 4: SINUSOIDAL MODELING LESSON 5: HORIZONTAL SHIFTS & SINUSOIDAL MODELING LESSON 6: THE RECIPROCAL TRIG FUNCTIONS This assignment is a teacher-modified version of Unit 11: The Circular Functions; Copright (c) 016 emath Instruction, LLC used b permission.
LESSON 1 BASIC GRAPHS OF SINE AND COSINE COMMON CORE ALGEBRA II The sine and cosine functions can be easil graphed b considering their values at the, those that are integer multiples of 90 or radians. Due to considerations from phsics and calculus, most trigonometric graphing is done with the input angle in units of!! General Form of sine and cosine equations: Eercise #1: Consider the functions f sin and g cos, where is an angle in radians. (a) B using the unit circle, fill out the table below for selected quadrantal angles. (b) Graph the sine curve on the grid shown below. 3 3
(c) B using the unit circle, fill out the table below for selected quadrantal angles. (d) Graph the cosine curve on the grid shown below. 3 3 (e) The domain and range of the sine and cosine functions are the same. State them below in interval notation. Domain: Range:
Recall: General Form of sine and cosine equations: b: frequenc: the number of curves in radians or 360 Complete sine curve (one ccle) Complete cosine curve (one ccle) period: the number of radians (or degrees) it takes for one complete curve (ccle) p = (f) After how much horizontal distance will both sine and cosine repeat its basic pattern? This is called the period of the trigonometric graph. Because these graphs have patterns that repeat the are called periodic. period = = * How can ou tell which is the sine curve and which is the cosine curve right awa?
Now we would like to eplore the effect of changing the coefficient of the trigonometric function. In essence we would like to look at the graphs of functions of the forms: = Asin(B) and = Acos(B) Amplitude = A The amplitude tells ou how the curve will go from the of the curve. It gives the range of the graph and can be stated as : - A A Consider the value of the amplitude in our first two graphs and where the middle of the curve was.. Eercise #: The grid below shows the graph of cos. Graph on the same set of aes. 3 3
Eercise #3: Graph 3 cos 3 3 Eercise #4: Graph cos 3 3 Eercise #5: The basic sine function is graphed below. Without the use of our calculator, sketch on the same aes. sin 3 3
Eercise #6: Graph 3sin 3 3 Eercise #7: Graph 1 sin 3 3 Eercise #8: Determine the amplitude and range for the following sine and cosine functions. (a) (b) (c)
Eercise #9: Determine the amplitude, frequenc, period, and range of the sine and cosine functions below; must be in radian form. (a) (b) (c)
FLUENCY BASIC GRAPHS OF SINE AND COSINE COMMON CORE ALGEBRA II LESSON 1 HOMEWORK Graph each of the following on the sets of aes provided. 1. sin 3 3. sin 3 3
3. 5sin 3 3 4..5cos 3 3
5. 3cos 3 3 7 sin 6. 3 3
7. Which of the following represents the range of the trigonometric function 7sin (1) 7, 7 (3) 0, 7 () 7, 7 (4) 7, 7 8. Which of the following is the period of cos (1) (3) () (4) 3?? 9. Which of the following equations describes the graph shown below? (1) 3cos () 3cos (3) 3sin 3 3 (4) 3sin 10. Which of the following equations represents the periodic curve shown below? (1) 4cos () 4cos (3) 4sin 3 3 (4) 4sin 11. Which of the following lines when drawn would not intersect the graph of 6sin (1) 8 (3) 4 () 3 (4) 9?
LESSON VERTICAL SHIFTING OF SINUSOIDAL GRAPHS COMMON CORE ALGEBRA II An graph primaril comprised of either the sine or cosine function is known as. These graphs can be stretched verticall, as we saw in the last lesson. Other transformations can occur as well. Toda we will eplore graphs of equations of the form: Since we alread understand the effect of A on the graph, it is now time to review the effect of adding a constant to an equation. Eercise #1: Consider the function f sin 3. (a) How would the graph of sin be shifted to produce the graph of f? 3 3 (b) On the grid to the right is the basic sine curve, sin grid, sketch the graph of. On the same f. For curves that have the general form, the value C is called the of the trigonometric function. It is the height or horizontal line that the sinusoidal curve rises and falls above and below b a distance of A (the amplitude). It is beneficial to sketch in the midline first before ou graph the sine or cosine function.
Eercise #: Consider the function cos 1. (a) Give the equation of a horizontal line that this curve rises and falls two units above. Sketch this line on the graph. 3 3 (b) Graph the equation. (c) State the range of this trigonometric function in interval notation. GRAPHING A SINUSOIDAL CURVE WITH A VERTICAL SHIFT 1. State the of the equation.. Use the of the function to plot points (pa attention to whether it s a sine or cosine curve) 3. Label the curve with the equation.
Eercise #3: Sketch and label the functions state the ranges of each of the equations in interval notation. 4sin and cos 3 on the grid below. Then, Range of 4sin : Range of cos 3: 3 3 Eercise #4: Determine the range of each of the following trigonometric functions. Epress our answer in interval notation. (a) 7sin 4 (b) 5cos (c) 5sin 35 RANGE:
Eercise #5: The graph below shows a sinusoidal curve of the form A and C. Show how ou arrived at our results. Asin C. Determine the values of, MIDLINE = C A = C + A =,6 Eercise #6:. The graph below shows a sinusoidal curve of the form of A and C. Show how ou arrived at our results.. Determine the values
FLUENCY VERTICAL SHIFTING OF SINUSOIDAL GRAPHS COMMON CORE ALGEBRA II LESSON HOMEWORK Sketch each of the following equations on the graph grids below. Label each with its equation. 1. 4sin 3 3. cos 4 3 3
3. sin 4 3 3 4. Graph and label both of the curves below. Then, state their intersection points (in other words, solve the sstem of equations shown below). 4cos 1 cos 4 3 3 Intersection Points:
5. The following graph can be described using an equation of the form cos of A and C. Show how ou arrived at our answers., 9 A C. Determine the values 0, 9, 9, 5, 5 6. The following graph can be described using an equation of the form sin of A and C. Show how ou arrived at our answers. 3, 30 A C. Determine the values, 30, 6 3, 6 7. State the range of each of the following sinusoidal functions in interval form. (a) 10sin 3 (b) 8cos (c) sin 30
8. When graphed, the line 14 would not intersect the graph of which of the following functions? (1) 5cos 9 (3) sin 15 () 6cos 10 (4) 3sin 0 9. Which of the following functions has a maimum value of 5? (1) 5sin 1 (3) 8cos 17 () 10cos 35 (4) 5sin 15
LESSON 3 THE FREQUENCY AND PERIOD OF A SINUSOIDAL GRAPH COMMON CORE ALGEBRA II A final transformation will allow us to stretch and compress sinusoidal graphs. It is important to be able to do this, especiall when modeling real-world phenomena, because most periodic functions do not have a period of. The first eercise will illustrate the pattern. Eercise #1: On the grid below is a graph of the function Graph. 3cos. How man full ccles of this function now fit within radians? 3 3 How are we going to fit these curves in the same amount of space? 3 3
The period, P, of a sinusoidal function is an etremel important concept. It is defined as. The period for the basic sinusoidal graphs is. Clearl, from our first eercise, the period of the function depends on the coefficient B in the general equations Asin B and Acos B. This coefficient, B, is known as the frequenc. Recall: period = = 1 Eercise #: Graph 3cos. 3 3 Clearl we can see from Eercises #1 and # that the frequenc and period are, that is as one increases the other decreases and vice versa. Eercise #3: Determine the period of each of the following sinusoidal functions. Epress our answers in eact form. (b) 8cos 3 (a) 6sin 4 (c) 1sin 3
Eercise #4: Sketch the function sin 4 on the grid below for one full period to the left and right of the -ais. Label the scale on our aes. Eercise #5: Sketch the function the -ais. Label the scale on our aes. on the grid below for one full period to the left and right of Eercise #6: Sketch the function of the -ais. Label the scale on our aes. on the grid below for one full period to the left and right
Eercise #7: On the aes below, graph one ccle of the cosine function with amplitude 3, period, midline = -1, and passing through the point (0,). Eercise #8: The heights of the tides can be described using a sinusoidal model of the form cos If high tides are separated b 4 hours, which of the following gives the frequenc, B, of the curve? (1) 1 (3) 1 A B C. () 4 (4) 6
Eercise #9: Determine an equation for this graph: Eercise # 10: Eercise #11: The graph shown below can be described using the equation = Acos(B) + k. Which of the following is the value of B + K? (1) 5π (3) 11 () 13 (4) π/7
FLUENCY THE FREQUENCY AND PERIOD OF A SINUSOIDAL GRAPH COMMON CORE ALGEBRA II LESSON 3 HOMEWORK 1. For each of the following sinusoidal functions, determine its period in eact terms of pi. (a) 6sin 10 (b) 1 7sin 3 (c) 4 cos 3 3 (d) 8sin 0.5. For each of the following sinusoidal functions, determine its eact period. (a) 5sin 7 (b) 5cos t1 365 (c) 8sin 1 9
3. If the period of a sinusoidal function is equal to 18, which of the following gives its frequenc? (1) 9 (3) 18 () 18 (4) 6 4. It is known for that a particular sine curve repeats its fundamental pattern after ever units along the - 7 ais. Which of the following is the frequenc of this curve? (1) 7 (3) 7 () 7 (4) 14 5. Determine an equation for this graph:
6. Determine an equation for this graph: 7. When the period of a sine function doubles, the frequenc (1) doubles. (3) is halved. () increases b. (4) decreases b. 8. Which of the following graphs shows the relationship between the frequenc, B, and the period, P, of a sinusoidal graph? Eperiment on our calculator. Graph the epression P. B P (1) (3) P B B P () (4) P B B
9. Sketch the function on the grid below for one full period to the left and right of the -ais. Label the scale on our aes. 10. Sketch the function on the grid below for one full period to the left and right of the -ais. Label the scale on our aes.
11. On the aes below, graph one ccle of the sine function with amplitude 4, period, midline =, and passing through the point.
1. Consider the curve whose equation is cos 3. 8 (a) Determine the eact period of this sinusoidal function. (b) What is the amplitude of this sinusoidal function? (c) What is the midline value of this sinusoidal function? (d) Sketch the function on the aes for a full period on both sides of the -ais. Label the scale on our -ais.
LESSON 4 SINUSOIDAL MODELING COMMON CORE ALGEBRA II The sine and cosine functions can be used to model a variet of real-world phenomena that are periodic, that is, the repeat in predictable patterns. The ke to constructing or interpreting a sinusoidal model is understanding the phsical meanings of the coefficients we ve eplored in the last three lessons. SINUSOIDAL MODEL COEFFICIENTS For sin and cos A B C A B C A C B P the amplitude or the sinusoidal model rises and falls above its the midline or of the sinusoidal model the frequenc of the sinusoidal model the amount of curves in radians the period of the sinusoidal model the minimum distance along the -ais for the ccle to Eercise #1: The tides in a particular ba can be modeled with an equation of the form cos d A Bt C, where t represents the number of hours since high-tide and d represents the depth of water in the ba. The maimum depth of water is 36 feet, the minimum depth is feet and high-tide is hit ever 1 hours. (a) On the aes, sketch a graph of this scenario for two full periods. Label the points on this curve that represent high and low tide. d (ft) (b) Determine the values of A, B, and C in the model. Verif our answers and sketch are correct on our calculator. t (hrs) (c) Tanker boats cannot be in the ba when the depth of water is less or equal to 5 feet. Set up an inequalit and solve it graphicall to determine all points in time, t, on the interval 0 t 4 when tankers cannot be in the ba. Round all times to the nearest tenth of an hour.
Eercise #: The height of a o-o above the ground can be well modeled using the equation h1.75cos t.5, where h represents the height of the o-o in feet above the ground and t represents time in seconds since the o-o was first dropped from its maimum height. (a) Determine the maimum and minimum heights that the o-o reaches above the ground. Show the calculations that lead to our answers. (b) How much time does it take for the o-o to return to the maimum height for the first time? Eercise #3: A Ferris wheel is constructed such that a person gets on the wheel at its lowest point, five feet above the ground, and reaches its highest point at 130 feet above the ground. The amount of time it takes to complete one full rotation is equal to 8 minutes. A person s vertical position,, can be modeled as a function of time in minutes since the boarded, t, b the equation cos A Bt C. Sketch a graph of a person s vertical position for one ccle and then determine the values of A, B, and C. Show the work needed to arrive at our answers. (ft) t (min)
Eercise #4: The possible hours of dalight in a given da is a function of the da of the ear. In Poughkeepsie, New York, the minimum hours of dalight (occurring on the Winter solstice) is equal to 9 hours and the maimum hours of dalight (occurring on the Summer solstice) is equal to 15 hours. If the hours of dalight can be modeled using a sinusoidal equation, what is the equation s ampltitude? (1) 6 (3) 3 () 1 (4) 4 Regents Question: Eercise #5: A sine function increasing through the origin can be used to model light waves. Violet light has a wavelength of 400 nanometers. Over which interval is the height of the wave decreasing, onl? (1) (0, 00) (3) (00, 400) () (100, 300) (4) (300, 400)
SINUSOIDAL MODELING COMMON CORE ALGEBRA II LESSON 4 HOMEWORK APPLICATIONS 1. A ball is attached to a spring, which is stretched and then let go. The height of the ball is given b the 4 sinusoidal equation 3.5cos t 5, where is the height above the ground in feet and t is the 5 number of seconds since the ball was released. (a) At what height was the ball released at? Show the calculation that leads to our answer. (d) Draw a rough sketch of one complete period of this curve below. Label maimum and minimum points. (ft) (b) What is the maimum height the ball reaches? (c) How man seconds does it take the ball to return to its original position? t (min)
. An athlete was having her blood pressure monitored during a workout. Doctors found that her maimum blood pressure, known as sstolic, was 110 and her minimum blood pressure, known as diastolic, was 70. If each heartbeat ccle takes 0.75 seconds, then determine a sinusoidal model, in the form sin A Bt C, for her blood pressure as a function of time t in seconds. Show the calculations that lead to our answer. 3. On a standard summer da in upstate New York, the temperature outside can be modeled using the sinusoidal equation Ot 11cos t 71, where t represents the number of hours since the peak 1 temperature for the da. (a) Sketch a graph of this function on the aes below for one da. 90 O (degrees F) (b) For 0 t 4, graphicall determine all points in time when the outside temperature is equal to 75 degrees. Round our answers to the nearest tenth of an hour. 50 t (hours) 4
4. The percentage of the moon s surface that is visible to a person standing on the Earth varies with the time since the moon was full. The moon passes through a full ccle in 8 das, from full moon to full moon. The maimum percentage of the moon s surface that is visible is 50%. Determine an equation, in the form P Acos Bt C for the percentage of the surface that is visible, P, as a function of the number of das, t, since the moon was full. Show the work that leads to the values of A, B, and C. 5. Evie is on a swing thinking about trigonometr (no seriousl!). She realizes that her height above the ground is a periodic function of time that can be modeled using h3cos t5, where t represents time in seconds. Which of the following is the range of Evie s heights? (1) h 8 (3) 3h 5 () 4h 8 (4) h 5
LESSON 5 HORIZONTAL SHIFTS & SINUSOIDAL MODELING HORIZONTAL SHIFTS: A in a sinusoidal graph is also known as a. This is when a constant is added to the function within the parentheses - that is, the constant must be added to the angle, not the whole function. The graph to the right is a shift of the sine function. The function is being shifted to the right π/ units. Phase Shifts of Sinusoidal Graphs: = sin( + C) = sin( - C) graph shifts C units graph shifts C units The same rules appl to cosine graphs.
Eamples 1.) Relative to the graph of = 3sin, what is the shift of the graph of = 3sin( + π/3)? (1) π/3 right () π/3 left (3) π/3 up (4) π/3 down.) What would happen to the cosine graph if the equation, = cos were changed to: = cos( - π/)? 3.) What would happen to the graph of = sin if it were changed to = sin( + π) - 4?
Modeling with Phase Shifts Modeling problems with phase shifts are done similarl to ones without phase shifts. 4.) The Ferris wheel at the landmark Nav Pier in Chicago takes 7 minutes to make one full rotation. The height, H, in feet above the ground of one of the si-person cars can be modeled b minimum height, in feet is, where t is time, in minutes. Using H(t) for one full rotation, this car s (1) 150 () 70 (3) 10 (4) 0 5.) The average dail temperature T (in degrees Fahrenheit) in Fairbanks, Alaska, is modeled b the equation, where time t is measured in months. Time t = 0 represents Januar 1. (a) What is the maimum average dail temperature in Fairbanks, Alaska? (b) During what month(s) will the temperature be 8 o F?
6.) A cit averages 14 hours of dalight in June, 10 in December, and 1 in both March and September. The number of dalight hours varies sinusoidall over a period of one ear. It can be modeled b the equation Januar is t = 0).. What is the average amount of dalight hours in August? (Hint: The month of 7.) A Ferris wheel can be modeled b a cosine graph. A student is comparing the different heights of the Ferris wheel during the ride. When eamining the graph, should the student focus on the amplitude, period, or midline? Eplain our answer.
LESSON 5 HORIZONTAL SHIFTS & SINUSOIDAL MODELING HOMEWORK 1.) Determine the shifts from the function = sin for the eamples below. (a) (b) (c).) Determine the shifts from the function = cos for the eamples below. (a) (b) (c) 3.) A group of students decided to stud the sinusoidal nature of tides. Values for the depths of the water level were recorded at various times. At t = hours low tide was recorded at a depth of 1.8 m. At t = 8 hours, high tide was recorded at a depth of 3.6 m. The function can be modeled b the equation: thousandth of a meter.. What is the depth of the water at t = 1 hours? Round to nearest
4.) A mass suspended from a spring is pulled down a distance of feet from its rest position. The mass is released at time t = 0 and allowed to oscillate. The mass returns to this position after 1 second. The equation used to modeling the movement is given b, where t is time in seconds and h(t) is height, in feet. At what two times within one ccle is the spring at 1.5 feet? Round to the nearest thousandth of a second. 5.) An object hangs from a spring in a stable position. The spring is being pulled downwards and the object begins to oscillate. A group of students are studing the movement of the spring. When eamining the graph of the spring, what does the period represent? What does the amplitude represent?
LESSON 6 THE RECIPROCAL TRIG FUNCTIONS COMMON CORE ALGEBRA II We have now seen three primar trigonometric functions, the sine, cosine, and tangent functions. Each of these can be defined in terms of either triangle or. For each of these functions, though, there eists what is known as a reciprocal function. Their definitions are shown below. THE OTHER FOUR TRIGONOMETRIC FUNCTIONS 1. SECANT:. COSECANT: 3. COTANGENT: Eercise #1: Which of the following is closest to the value of sec5? (1) 0.6 (3) 0.36 () 1.6 (4).48 Eercise #: Considering our work with sine and cosine, evaluate each of the following. Epress our answers in eact and simplest form. (a) sec60 (b) cot 150 (c) 3 csc 4
Because each of these reciprocal trigonometric functions has a variable denominator, there will be angles at which these denominators are zero and hence the function is undefined. Eercise #3: Which of the following values of is not in the domain of csc? (1) 180 (3) 90 () 60 (4) 135 Unit Circle Chart:
Because each of these functions is dependent on sine and/or cosine, it is possible to determine the sign (positive or negative nature) of each based on the quadrant of the input angle. Eercise #4: Determine the sign of each of the following trigonometric functions in the quadrant specified. (a) cot for in quad. II (b) sec for in quad. IV (c) csc for in quad. III Eercise #5: If cot 0 and sec 0 then could be which of the following angles? (1) 48 (3) 1 () 310 (4) 5
We should also be able to produce all of the trigonometric ratios (all SIX of them) if we are given a right triangle. Eercise #6: A right triangle is shown below with sides 3 and 4. State the value of each of the following trigonometric ratios. sin A csc A B cos A sec A 3 tan A cot A C 4 A Eercise #7: If is an angle whose terminal ra lies in the fourth quadrant and eact value of csc. Show how ou arrived at our answer. 1 cos, then determine the 3 Eercise #8: 11. The angle when drawn in standard position has its terminal ra in the second quadrant. If it known that, then determine each of the following values. a) b) c)
THE RECIPROCAL TRIG FUNCTIONS COMMON CORE ALGEBRA II LESSON 11 HOMEWORK FLUENCY 1. Determine the value of each of the following in eact and simplest form (leave no comple fractions). (a) csc30 (b) cot 90 (c) sec180 (d) cot 3 (e) 3 csc (f) 5 sec 4. Use our calculator to determine the value of each of the following to the nearest hundredth. (a) cot 115 (b) sec31 (c) csc45 3. In simplest radical form, sec135 is equal to (1) (3) 3 () (4) 3
4. Which of the following is nearest to the value of cot 0? (1) 1.19 (3).74 () 3.17 (4) 0.85 5. For which of the following values of is cot undefined? (1) 60 (3) 180 () 90 (4) 135 6. For which angle,, below will sec not eist? (1) 30 (3) 180 () 45 (4) 90
7. For the angle it is known that csc 0 and terminal ra of lies in quadrant (1) I (3) III () II (4) IV sec 0. When drawn in standard position, the 8. The angle when drawn in standard position has its terminal ra in the second quadrant. If it is known 5 that sin then determine the values of all of the remaining trigonometric functions. 13 (a) cos (b) tan (c) sec (d) csc (e) cot