Precalculus Honors Problem Set: Elementary Trigonometry

Transcription

1 Precalculus Honors Problem Set: Elementary Trigonometry Mr. Warkentin 03 Sprague Hall Academic Year Directions: These questions are not presented in order of difficulty. Some of these questions are quick, some long. Real learning should happen when you are doing many of these homework problems - stick with questions, wrestle with them, and you will be rewarded with a much stronger understanding of our content. Unless otherwise indicated, your answers must be exact or have four significant figures. Questions labeled Challenge are genuinely tough questions that will improve your mental discipline, creativity, and persistence. While you are not required to do these questions, students considering a STEM major should at least make an attempt. Any extra credit problems in this class will be similar to challenge problems. 1. Recall from class that the "transformation form" of function f is the following: a f (b(x h)) + k. Quickly describe, in words, the types of transformations that are effected by the undetermined constants a, b, h, and k. Which of these relate to period of a sinusoid? Amplitude? Frequency? Phase shift? a does vertical shrink/stretch/reflection. b does horizontal shrink/stretch/reflection. h does horizontal translation. k does vertical translation. a related to amplitude. b relates to period and frequency. h relates to phase shift. k relates to none of the terms.. Give two values for the angle θ (in radians) which satisfies the equality csc θ =. θ = π 6, 5π 6, or any angle coterminal with these two. 3. Give two values for the angle A (in radians) which satisfies the equality cota = 3. θ = 5π 6, π 6, or any angle coterminal with these two. 4. Which of the 1 basic functions from the textbook section 1.3 are bounded above? below? both? Bounded above: none are only bounded above Bounded below: squaring, square root, exponential, absolute value Bounded above and below: sine, cosine, logistic

2 5. Consider the following function r(x) = x +. We want to apply the following transformations to this function: a horizontal stretch by a factor of, a vertical shift of 1 unit down, and a horizontal reflection. What will be the equation of the transformed function? In this case, a = 1, b = 1, h = 0, and k = 1. Next we need to plug into transformation form: r 1 x 1 and hence, by the tool of composition, the resulting expression is (noting that the 1 is not under the square root): 1 x Avery is riding a Ferris wheel that turns once every 4 seconds, and which has a radius of 8 meters. The function h defined by h(t) = 9 8 cos πt 1 describes Avery s distance from the ground (in meters) after t seconds of riding. For example, h(8) = 13 means that Avery is 13 meters above the ground after 8 seconds of riding. a. Evaluate h(0), and explain its significance. b. Explain why h(16) = h(8). c. Find a value for t that fits the equation h(t) = 10. What is the meaning of this calculation? d. Explain why h(t + 4) = h(t) is true, no matter what value t has. e. What are all the possible outputs of the function h(t)? Adapted from a problem set published by Phillips Exeter Academy a. h(0) = 1... this means that Avery starts at the bottom of the ferris wheel, a height of 1 meter. b. Because if we go the same distance to the right or left of a peak, we ll still be at the same height. Avery reaches a peak at t = 1, so her height will be the same at t = 16 and t = 8. c. It is impossible for us to calculate an exact value. However, we know that (by algebra) cos πt 1 = 1 8. We can fiddle around with values of t... we could choose πt 1 to be very close to π, since cos π = 0 and 0 is very close to 1 8. After calculating an appropriate t-value, we have found a time at which Avery is 10 meters above the ground. d. h is a 4-periodic function. e. Its range is all real numbers between 1 and [1, 17] 7. Find all six trigonometric functions (sine, cosine, tangent, secant, cosecant, and cotangent) of the following angles by hand. Feel free to check your work with a calculator (being careful, as always, to ensure that it is in the proper mode). a. 405

3 b. 13π rad c. 10 d. 13π 3 rad a. θ = cos (θ) =, sin (θ) =, tan (θ) = 1, sec (θ) =, csc (θ) =, cot (θ) = 1 b. θ = 13π rad - cos (θ) = 0, sin (θ) = 1, tan (θ) = undefined, sec (θ) = undefined, csc (θ) = 1, cot (θ) = 0 c. θ = 10 - cos (θ) = 3, sin (θ) = 1, tan (θ) = 1 3, sec (θ) = 3, csc (θ) =, cot (θ) = 1 3 d. θ = 13π 3 rad - cos (θ) = 1, sin (θ) = 0, tan (θ) = 0, sec (θ) = 1, csc (θ) = undefined, cot (θ) = undefined 8. Convert the following functions into a transformation form of the given elementary function (calculate the values of a, b, h, and k then plug them into the form): a. Write f (x) = 10 [x + 4] as a transformation of the elementary function x. b. Write g(x) = ln x as a transformation of the elementary function ln(x). c. Write (x) = 1.5 cos πx π as a transformation of the elementary function cos(x). What is this sinusoid s period, amplitude, frequency, and phase shift? a. Write f (x) = 10 [x + 4] as a transformation of the elementary function x. a = 10, b =, h =, k = 0... so 10 ([x ( )]) is the answer. b. Write g(x) = ln x as a transformation of the elementary function ln(x). a = 1, b = 1, h = 6, k = 1... so ln 1 [x 6] + 1 is the answer. c. Write (x) = 1.5 cos πx π as a transformation of the elementary function cos(x). What is this sinusoid s period, amplitude, frequency, and phase shift? a = 1.5, b = π, h = 1, k = 0... so 1.5 cos π x 1 is the answer. The period is, the amplitude is 1.5, the frequency is 1, and the phase shift is 1 unit to the left. 9. Using your newfound knowledge of transformations, easily sketch the graphs of the following functions. In this course, sketching means plotting and labeling a few points, then using your knowledge of the function s behavior to imprecisely draw the rest of the curve. After sketching, check your work with a graphing calculator. a. f (x) = 10 [x + 4] b. g(x) = ln x c. (x) = 1.5 cos πx π

4 d. A sine wave with an amplitude of 10, a vertical shift up by 5, and a peak at x = π Just check it with Desmos Look at the pictures below. Write the expressions for two functions which generate each graph. (In other words, you will have four functions in your answer). How can you easily create as many functions as you would like for each of the graphs? Possible answers for graph 1: 3 sin(πx) and 3 cos π x 1 Possible answers for graph : 1.5 cos 1 x + π and 1.5 sin 1 x 5π 11. Look at the following picture. At 67 meters in height, the "Big O" ferris wheel in Tokyo is the largest centerless ferris wheel in the world.

5 a. Which function should be used to model just the height of a cabin as the wheel rotates? b. If the wheel rotates once every 0 minutes (and assuming the bottom of the wheel is barely above the ground), write a function which models the height of a cabin. c. There are multiple correct answers to both parts a and b; why? How many possible answers are there? a. Sine, since it is defined to be height on the unit circle. b. h(t) = 33.5 sin π 10 t Given the wording of the question, you do not need a phase shift. However, if you want h(0) = 0, then we must include a phase shift of 5. c. Approximately infinity - they re all just shifts of each other. 1. Every time a person s heart beats, blood pressure increases and then decreases. The average person s heart beats once every 0.86 seconds, and the average person has a maximum pressure of 10mmHg and a minimum of 80mmHg (where mmhg, millimeters of mercury, is a unit of pressure). Given these facts, write an expression for a sinusoidal function, P(t), that models an average person s blood pressure over time (where t is measured in seconds) if the heart is at rest (lowest pressure) at time t = 0. P(t) = 0 sin π 0.86 (t 0.15) Alternatively, you could take a vertical reflection of cosine (in order to get a valley along the y-axis) as follows: P(t) = 0 cos π 0.86 t Consider the graph of a sinusoid which has not been shifted vertically (i.e. a sine wave which wiggles an equal distance above and below the x-axis). If you are given a peak and a valley of the function, is there necessarily a zero between them? Why or why not? If yes, how can we find the zero? If no, sketch the graph of a sinusoid which demonstrates this. Yes, we can find the zero at the x-coordinate precisely between the peak and valley s x-coordinates.

6 14. Find an angle θ 1 such that sin(θ 1 ) = csc(θ 1 ), or show that such an angle does not exist (DNE). θ 1 = π, θ 1 = π, or any angle coterminal with those listed. 15. Find an angle θ such that sec(θ ) = tan(θ ), or show that such an angle does not exist (DNE). No such angle exists because: sec(θ ) = tan(θ ) 1 cos(θ ) = sin(θ ) cos(θ ) sin(θ ) = cos(θ ) cos(θ ) by the definitions of secant and tangent However, this is impossible since, if the sine of an angle is 1, then the cosine of that angle must be 0. If the cosine is 0, then tangent doesn t exist. Thus, there is no workable angle. 16. Where will the graph of the function tan x sec x have asymptotes? Why? No asymptotes because tan x sec x = sin x cos x 1 cos x = sin x cos x cos x The graph of this function will look exactly like the graph of the sine funtion, but it will have holes because it is undefined anywhere cos x = 0. We will explore this property more later in the year. 17. a. Consider the following expression: x π. For which value of x does x π = 0? For which value of x does x π = π? x π = π? b. Now consider the following function l: l(x) = 1 cos(x π). Take your x-values to part (a) and plug them into l(x). Describe and explain the results. c. How do your answers to parts (a) and (b) relate to the graph of l? d. Use this technique to find 5 zeroes of l. e. What are the coordinates of peaks and valleys on the graph of l? a. If x = π then x π = 0. If x = π then x π = π. If x = 3π then x π = π. b. Sequentially, they give us 1, 1, and 1 - these are the peaks and valleys of function l. This should make sense because we are, essentially, plugging in 0, π, and π into the cosine function. c. See previous answer. d. We need x π = π, so x = 3π 4 by algebra. So x = 3π 4 is one zero of l. Now we just add/subtract the half-period from that value (since zeroes of sinusoids are separated by half of a period). Thus, we know that 3π 4, π 4, π 4, 3π 4, and 5π 4 are all zeroes of l.

7 e. Just add the full period (π) to one of the peaks or valleys we found earlier in order to find the x-coordinate of a peak or valley, respectively. 18. Look at the pictures below. Write the expressions for two functions which generate each graph. (In other words, you will have four functions in your answer). Possible answers for graph 1: 3 cot x and 3 tan x π Possible answers for graph : 1 csc(πx) and 1 sec π x Sketch the graphs of the following functions. After sketching, check your work with a graphing calculator.

8 a. (x) = sec x + π b. (x) = 0 cot x Just check with Desmos Challenge: A wheel of radius 1 meter is centered at the origin, and a rod AB of length 3 meters is attached at A to the rim of the wheel. The wheel is turning counterclockwise, one rotation every 4 seconds, and, as it turns, the other end B = (b, 0) of the rod is sliding back and forth along the x-axis. Given any time t seconds, the coordinate b is determined by the expression b = f (t). The picture below shows this situation when t = Challenge: Look at the graph of the infinitiely wiggly function sin 1 x and zoom in really far. Explain in words why this function behaves this way.. Calculate by hand (showing your algebra) at least three zeroes of the function f (x) = 3 cos π (x 5). How do these zeroes relate to the graph of the function y = 3 sec π (x 5)? The zeroes of this function are all even integers. They are the vertical asymptotes of the secant function. 3. Let f (x) = sin(x) cos(x). a. Is f a periodic function? If so, what is its period? b. Is f a bounded function? If so, what are its bounds? c. Do you think that f is sinusoidal? Explain why or why not. Adapted from a problem set published by Phillips Exeter Academy a. π-periodic because sin(x + π) cos(x + π) = ( sin x)( cos x) = sin x cos x. b. This is a much harder question, but given the symmetries of this problem it s a reasonable guess that upper and lower bounds will be achieve at odd multiples of π 4, where f achieves ± 1.

9 c. It is sinusoidal if we can find a, b, h, and k values of a sinusoid that give us the same function - sin(x) works. 1 Adapted from a problem set published by Phillips Exeter Academy 4. State two sinusoidal functions which, when added together, have the same graph as cos(x π). The lazy mathematician s answer is 1 cos(x π) + 1 cos(x π), but there are approximately answers to this question. 5. For each of the following functions, decide whether it is periodic on its domain. If it is, state its period. Recall that period is defined to be the smallest positive number p such that f (x +p) = f (x) for any x-value in the function s domain. a. f 0 (x) = 4 cos(10x) + x b. f 1 (x) = 4 cos(10x) + 1 c. f (x) = tan(πx) d. For function f 3, look at the following picture. e. f 4 (x) = (sec x + tan x) f. f 5 (x) = sin(πx) + sin x a. Non-periodic π b. 5 -periodic c. 1-periodic d. π-periodic e. π-periodic (Desmos is important on this one.) f. Non-periodic because π and 1 do not have a least common multiple (LCM). 6. The term resonance refers to two overlapping sine waves which reinforce each other s amplitudes. In other words, when the two sine waves are combined, their amplitude is higher than each of their individual amplitudes. Similarly, interference refers to a canceling effect when two sine waves are combined. Consider the following function: ω(θ) = sin (b (x h 1 )) + cos (b (x h )) We have already seen that adjusting the value of b determines whether or not the sum is a sinusoid. So, let s just assume that the period is the same (pick a value - perhaps 1, perhaps π). How do the values of

10 h 1 and h (the phase shifts of the two sine functions) influence whether we see resonance or interference? Desmos or a graphing calculator is necessary for this question. If the waves are in-phase (that means their phase shifts lead to peaks and valleys close to each other), there is resonance. If the valleys are close to peaks and vice versa, then there is interference. 7. Challenge: We have defined a p-periodic function to be a function such that f (x + p) = f (x). Now, let us similarly define a p-antiperiodic function to be one such that f (x + p) = f (x). Is it possible for a function to be both periodic and antiperiodic? Prove your answer with algebra either way. Then, sketch the graph of an antiperiodic function, labeling multiple points to demonstrate its antiperiodicity. 8. Challenge Let f (x) be an odd function and let g(x) be even. Consider the function Prove that h is an odd function. h(x) = f (x) g(x) + f (x). Honor Code In the space below, please write "I have neither given nor received unauthorized aid on this work." and sign your name.