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Chapter 4 Lesson Vocabulary The Tangent Function and Periodicity 4-6 tangent function periodic function period of a function BIG IDEA The sine and cosine functions are periodic, repeating every 2π or 360º. The tangent function is periodic, repeating every π or 180º. A frieze pattern is a visual design that repeats over and over along a line. The frieze pattern at the right appears on the Chan Chan ruins in Trujillo, Peru. Mental Math How many times does the minute hand of a clock pass the number 6 between 10 A.M. and 6 P.M.? In Lesson 4-5, you used values of sine and cosine to graph trigonometric functions. You also observed that, like frieze patterns, their graphs repeat as you move horizontally. This lesson extends those ideas to the tangent function. The Tangent Function The correspondence θ → tan θ, when θ is a real number, defines the sin θ tangent function. From the definition tan θ = _____ , values for the cos θ tangent function can be generated. Activity Step 1 The table below contains some exact values of tan θ. It also shows decimal equivalents of those values. Fill in the missing values. 252 θ 0 π 30º = __ π 45º = __ π 60º = __ tan θ (exact) 0 √ 3 ___ 3 1 √ 3 undefined ? ? ? 0 tan θ (approx.) 0 0.577 1 1.732 undefined ? ? ? 0 6 4 π 120º = ___ 2π 3π 5π 90º = __ 135º = ___ 150º = ___ 180º = π 4 3 6 3 2 θ 7π 210º = ___ 6 5π 225º = ___ 4 4π 240º = ___ 3 3π 270º = ___ 2 5π 300º = ___ 3 7π 11π 315º = ___ 330º = ____ 360º = 2π 4 6 tan θ (exact) ? ? ? ? ? ? ? ? tan θ (approx.) ? ? ? ? ? ? ? ? Trigonometric Functions SMP_SEFST_C04L06_252_256_FINAL.i252 252 4/30/09 3:56:22 PM Lesson 4-6 Step 2 At the right, a graph of the values of the tangent function given in the first part of the table from Step 1 is shown. Copy this graph, and add the points you found in Step 1 to the graph. Step 3 Draw a smooth curve through these points, to show the graph of y = tan θ for all θ, 0º ≤ θ ≤ 360º, 0 ≤ θ ≤ 2π where tan θ is defined. y 2 1 θ 90˚ 180˚ 270˚ 360˚ 450˚ -1 -2 The Graph of the Tangent Function 3π 5π At the right is a graph of y = tan x for – ___ ≤ x ≤ ___ . Notice that this 2 2 graph looks strikingly different from the graphs of both the sine and cosine functions. The tangent function has asymptotes and does not have a maximum or minimum value. y h(x) = tan x x -270˚ -90˚ - 3π -π - π 2 2 90˚ π 2 270˚ 450˚ 2π 5π π 3π 2 2 Example 1 Consider f(x) = tan x. a. Give the domain and range of the function f. b. Is f an odd function, an even function, or neither? Justify your answer. Solution a. Because the tangent function has multiple vertical asymptotes, the domain of the tangent function is the set of all real numbers π except odd multiples of 90º or __ . Notice that the tangent function 2 has no minimum or maximum values. Therefore, its range is the set of all real numbers. b. From the Opposites Theorem, tan(–x) = –tan x for all x. Thus, the tangent function is an odd function. Periodicity and the Trigonometric Functions The periodic nature displayed by sine, cosine, and tangent is summarized in the following theorem. Periodicity Theorem For any real number x and any integer n, sin x = sin (x + n · 2π) = sin (x + n · 360º) cos x = cos (x + n · 2π) = cos (x + n · 360º) tan x = tan (x + n · π) = tan (x + n · 180º). The Tangent Function and Periodicity SMP_SEFST_C04L06_252_256_FINAL.i253 253 253 4/28/09 2:49:49 PM SMP_SEFST_C04L06_252_256_FINAL.i254 Page 254 4/29/10 1:07:44 PM other /Volumes/108/WG00060_r1/work%0/indd%0/SMP_FST_SE_...... Chapter 4 The theorem states that the sine and cosine functions are periodic functions with period 360º or 2π radians, while the tangent function is a periodic function with period 180o or π radians. Definitions of Periodic and Period A function f is periodic if there is a positive number p such that f(x + p) = f(x) for all x in the domain of f. The smallest such p, if it exists, is called the period of f. A part of the function from any particular x to x + p, where p is the period of the function, is called a cycle of the function. For instance, one π π cycle of the tangent function is from 0 to π; another is from – __ to __ . 2 2 Example 2 Use the Periodicity Theorem to find cos 2670º. 2670º Solution _____ 360º ≈ 7.4, so 2670º – 7 · 360º will be less than 360º. 2670º – 7 · 360º = 150º, so R2670º = R150º. Therefore, √ 3 cos 2670º = cos 150º = – ___ . 2 Many phenomena are periodic, including tides, calendars, heart beats, actions of circular gears, phases of the moon, and seasons of the year. GUIDED Example 3 a. the maximum and minimum values b. the range c. the period Blood Pressure y Pressure (mm mercury) The graph at the right shows normal human blood pressure as a function of time. Blood pressure is systolic when the heart is contracting and diastolic when the heart is expanding. The changes from systolic to diastolic blood pressure create the pulse. For this function, determine each. Systolic 130 120 110 100 90 0 11.6 Solution a. The maximum and minimum values of the graph are those values in which the graph obtains a highest and lowest point, respectively. The maximum value on this graph is ? , the minimum value is ? . b. The range is the maximum value minus the minimum value. From Part a, the range shown on this graph is ? . c. The period is the range of x-values for the smallest section of the graph that can be translated horizontally onto itself. The period shown on this graph is ? seconds. 254 Trigonometric Functions x Diastolic 11.8 12 12.2 Time (seconds) 12.4 SMP_SEFST_C04L06_252_256_FINAL.i255 Page 255 1/9/10 4:18:59 PM /Volumes/121/WG00060/work%0/indd%0/SMP_FST_SE_C04/SMP_SEFST_C04L06_252_ u-s082 Lesson 4-6 Questions COVERING THE IDEAS 1. a. List all values of θ between 0º and 360º such that cos θ = 0. b. What is f(θ) = tan θ for these θ-values? c. What do these values of θ mean for the graph of the tangent function? 2. List all of the values of x from 0 to 2π for which sin x = 0. What do these x-values indicate for the graph of the tangent function? In 3–5, use the Periodicity Theorem to evaluate. 3. sin 495º 4. cos 810º 5. tan 3570º 4π 6. Given that tan ___ ≈ 5.671, use the Periodicity Theorem to evaluate. 9 13π 5π 22π a. tan ____ b. tan – ___ c. tan ____ 9 9 9 7. What is the period of the function with the given equation? a. y = sin x b. y = cos x c. y = tan x APPLYING THE MATHEMATICS 8. Suppose that f is a periodic function whose domain is the real 6 numbers. One cycle of f is graphed at the right. a. What is the period of f ? y 4 f 2 b. Graph f on the interval –15 ≤ x ≤ 15. c. Find f(51). x -5 d. Find four integer values of x such that f(x) = 0. 5 -2 9. If one endpoint of a cycle of the cosine function is 90º, where is the other endpoint? π 10. If one endpoint of a cycle of the tangent function is __ , where is the 2 other endpoint? 11. State equations for two of the asymptotes of the tangent function a. in radians. b. in degrees. 12. Let f(n) be the number in the nth decimal place of __17 . a. Give the values of f(1), f(2), f(3), and f(4). b. f is a periodic function. What is its period? 13. The table at the right contains hourly data for the height of tide relative to the mean low water level in Pago Pago, American Samoa on October 5, 2008. a. Create a scatterplot of the data. b. Determine the range of the data. c. From the scatterplot, estimate the period of the data. Hour Height Hour Height 0 1.59 12 2.03 1 1.29 13 1.65 2 0.91 14 1.14 3 0.61 15 0.83 4 0.65 16 0.62 5 0.79 17 0.49 6 1.04 18 0.53 7 1.32 19 0.87 8 1.75 20 1.15 9 2.01 21 1.49 10 2.20 22 1.77 11 2.21 23 1.87 Source: National Oceanographic and Atmospheric Administration The Tangent Function and Periodicity 255 SMP_SEFST_C04L06_252_256_FINAL.i256 Page 256 1/9/10 4:19:07 PM /Volumes/121/WG00060/work%0/indd%0/SMP_FST_SE_C04/SMP_SEFST_C04L06_252_ u-s082 Chapter 4 REVIEW 14. Fill in the blanks for the graph of the sine function at the right. (Lesson 4-5) 15. Refer to the predator-prey graph below. (Lesson 4-5) ( ? , 0) (?, ?) Population For both the predator and the prey functions, determine the a. domain. b. maximum and minimum. c. period. 1200 1100 1000 900 800 700 600 500 400 300 200 100 x (3π, 0) ( ? , 0) Prey Predator 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 Months √ 3 16. Find x such that 0 ≤ x ≤ 2π, if cos x = __12 and sin x = – ___ . 2 (Lesson 4-4) 17. Is the cosine function odd, even, or neither? Justify your conclusion. (Lessons 4-5, 3-4) 18. Suppose Rθ(1, 0) = (-0.75, y) and is a point in Quadrant II. Find y. (Lesson 4-3) 19. Under some translation T, the point (–6, 2) is mapped to (0, 7). a. State the rule for T. b. Find T(9, 9). (Lesson 3-2) 20. The graph at the right is from Weather on T 40 Temperature (˚C) the Planets, by George Ohring. It shows how temperature is a function of latitude on Earth and Mars when it is spring in one hemisphere on each planet. Let L be the latitude on each planet, E(L) be the average temperature at latitude L on Earth, and M(L) be the average temperature at latitude L on Mars. a. Is L the dependent or independent variable? b. Estimate E(60). c. Estimate E(0) - M(0), and state what quantity this expression represents. d. What is the range of M? (Lesson 2-1) 20 Trigonometric Functions Mars Average T = M(L) 0 -40 -60 -80 L -100 90 60 30 0 30 60 North Pole 21. The word tangent has another meaning in geometry. It also has 256 T = E(L) -20 EXPLORATION another meaning in English. What are these meanings? Earth Average 90 South Pole Latitude (Degrees)