The Tangent Function and Periodicity 253

Chapter 4

Lesson The Tangent Function Vocabulary 4-6 tangent function and Periodicity 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 Mental Math a line. The frieze pattern at the right appears on the Chan Chan How many times does the ruins in Trujillo, Peru. minute hand of a clock pass the number 6 In Lesson 4-5, you used values between 10 A.M. and of sine and cosine to graph 6 P.M.? 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, deﬁ nes the sin θ tangent function. From the deﬁ nition 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.

π π π π 2π 3π 5π θ 0 30º __ 45º __ 60º __ 90º __ 120º ___ 135º ___ 150º ___ 180º = 6 = 4 = 3 = 2 = 3 = 4 = 6 = π tan θ √3 0___ 1 √3 undeﬁ ned ???0 (exact) 3

tan θ 0 0.577 1 1.732 undeﬁ ned ???0 (approx.)

7π 5π 4π 3π 5π 7π 11π θ 210º ___ 225º ___ 240º ___ 270º ___ 300º ___ 315º ___ 330º ____ 360º 2 = 6 = 4 = 3 = 2 = 3 = 4 = 6 = π tan θ ???????? (exact) tan θ ???????? (approx.)

252 Trigonometric Functions

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y Step 2 At the right, a graph of the values of the 2 tangent function given in the fi rst part of the table from Step 1 is shown. Copy this graph, and add the points you found in 1 Step 1 to the graph. θ Step 3 Draw a smooth curve through these points, 90˚ 180˚ 270˚ 360˚ 450˚ to show the graph of y = tan θ for all θ, 0º ≤ θ ≤ 360º, 0 ≤ θ ≤ 2π where -1 tan θ is defi ned.

-2

The Graph of the Tangent Function y 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 h(x) = tan x have a maximum or minimum value. x -270˚ -90˚ 90˚ 270˚ 450˚ - 3π - - π π 3π 2 5π 2 π 2 2 π 2 π 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 253

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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.

Deﬁ nitions 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 ; another is from __ to __ . π – 2 2

Example 2 Use the Periodicity Theorem to fi nd cos 2670º. _____2670º Solution 360º ≈ 7.4, so 2670º – 7 · 360º will be less than 360º. Therefore, 2670º – 7 · 360º = 150º, so R2670º = R150º. √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 The graph at the right shows normal human blood Blood Pressure y pressure as a function of time. Blood pressure is Systolic systolic when the heart is contracting and diastolic 130 when the heart is expanding. The changes from 120 systolic to diastolic blood pressure create the pulse. 110 For this function, determine each. 100 Pressure 90 a. the maximum and minimum values (mm mercury) x 0 Diastolic b. the range 11.6 11.8 12 12.2 12.4 c. the period Time (seconds) 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.

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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 y 8. Suppose that f is a periodic function whose domain is the real 6 numbers. One cycle of f is graphed at the right. 4 a. What is the period of f ? f b. Graph f on the interval –15 ≤ x ≤ 15. 2 c. Find f(51). x -5 5 d. Find four integer values of x such that f(x) = 0. -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 Hour Height Hour Height a. in radians. b. in degrees. 0 1.59 12 2.03 1 12. Let f(n) be the number in the nth decimal place of __ . 7 1 1.29 13 1.65 a. Give the values of f(1), f(2), f(3), and f(4). 2 0.91 14 1.14 b. f is a periodic function. What is its period? 3 0.61 15 0.83 13. The table at the right contains hourly data for the height of tide 4 0.65 16 0.62 relative to the mean low water level in Pago Pago, American Samoa 5 0.79 17 0.49 on October 5, 2008. 6 1.04 18 0.53 a. Create a scatterplot of the data. 7 1.32 19 0.87 b. Determine the range of the data. 8 1.75 20 1.15 c. From the scatterplot, estimate the period of the data. 9 2.01 21 1.49 10 2.20 22 1.77 11 2.21 23 1.87 Source: National Oceanographic and Atmospheric Administration

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Chapter 4

REVIEW 14. Fill in the blanks for the graph of the sine function at the right. (Lesson 4-5) (3π, 0) x ( ? , 0) ( ? , 0) 15. Refer to the predator-prey graph below. (Lesson 4-5) ( ? , ? ) For both the predator and the prey functions, determine the a. domain. b. maximum and minimum. c. period.

1200 1100 Prey 1000 900 800 700 600 500

Population Predator 400 300 200 100

224 6 8 10 12 14 16 18 20224 26 28 30 32 34 36 38 40 42 44 46 48 Months

1 √3 16. Find x such that 0 x 2π, if cos x __ and sin x – ___ . ≤ ≤ = 2 = 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 the Planets, by George Ohring. It shows how 40 T = E(L) temperature is a function of latitude on Earth 20 Earth Average C)

and Mars when it is spring in one hemisphere ˚ 0 on each planet. Let L be the latitude on each -20 planet, E(L) be the average temperature at -40 latitude L on Earth, and M(L) be the average -60 Mars Average T = M(L) temperature at latitude L on Mars. -80 Temperature ( Temperature a. Is L the dependent or independent variable? L -100 b. Estimate E(60). 90 60 30 0 30 60 90 c. Estimate E(0) - M(0), and state what North Pole South Pole quantity this expression represents. Latitude (Degrees) d. What is the range of M? (Lesson 2-1) EXPLORATION 21. The word tangent has another meaning in geometry. It also has another meaning in English. What are these meanings?

256 Trigonometric Functions