Unit 6 Introduction to Trigonometry Graphing Other Trig Functions (Unit 6.5)

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1 Unit 6 Introduction to Trigonometry Graphing Other Trig Functions (Unit 6.5) William (Bill) Finch Mathematics Department Denton High School

2 Lesson Goals When you have completed this lesson you will: Graph the parent tangent, cotangent, secant, and cosecant funtions. Graph transformations of these functions. Graph damped trigonometric functions. Graph Other Trig Functions 2 / 22

3 Lesson Goals When you have completed this lesson you will: Graph the parent tangent, cotangent, secant, and cosecant funtions. Graph transformations of these functions. Graph damped trigonometric functions. Graph Other Trig Functions 2 / 22

4 Lesson Goals When you have completed this lesson you will: Graph the parent tangent, cotangent, secant, and cosecant funtions. Graph transformations of these functions. Graph damped trigonometric functions. Graph Other Trig Functions 2 / 22

5 Lesson Goals When you have completed this lesson you will: Graph the parent tangent, cotangent, secant, and cosecant funtions. Graph transformations of these functions. Graph damped trigonometric functions. Graph Other Trig Functions 2 / 22

6 Overview The tangent and cotangent functions. Graph Other Trig Functions 3 / 22

7 Overview The secant and cosecant functions. Graph Other Trig Functions 4 / 22

8 The Parent Tangent Function Domain: x R, x π 2 + nπ Range: (, ) x-intercept: nπ y-intercept: 0 Continuity: inf. discont. at x = π 2 + nπ Asymptotes: x = π 2 + nπ Symmetry: origin (odd function) Extrema: none End behavior: does not exist Graph Other Trig Functions 5 / 22

9 Period of the Tangent Function One period of the tangent function is π. For y = a tan(bx + c) + d the period is period = π b Graph Other Trig Functions 6 / 22

10 Example 1 Horizontal Dilation of Tangent Locate the vertical asymptotes and then sketch the graph of y = tan π 3 x. Graph Other Trig Functions 7 / 22

11 Example 2 Reflections and Translations of Tangent Locate the vertical asymptotes and sketch the graph of y = tan π 4 x Graph Other Trig Functions 8 / 22

12 Example 3 Reflections and Translations of Tangent Locate the( vertical asymptotes and sketch the graph of y = tan x + π ) 2 Graph Other Trig Functions 9 / 22

13 The Parent Cotangent Function Domain: x R, x nπ Range: (, ) x-intercept: π 2 + nπ y-intercept: none Continuity: inf. discont. at x = π 2 + nπ Asymptotes: x = nπ Symmetry: origin (odd function) Extrema: none End behavior: does not exist Graph Other Trig Functions 10 / 22

14 Period of the Cotangent Function One period of the cotangent function is π. For y = a cot(bx + c) + d the period is period = π b Graph Other Trig Functions 11 / 22

15 Example 4 Locate the vertical asymptotes and sketch the graph of y = cot 2x. Graph Other Trig Functions 12 / 22

16 The Parent Secant Function Domain: x R, x π 2 + nπ Range: (, 1] [1, ) x-intercept: none y-intercept: 1 Continuity: inf. discont. at x = π 2 + nπ Asymptotes: x = ı 2 + nπ Symmetry: y-axis (even function) End behavior: does not exist Graph Other Trig Functions 13 / 22

17 The Parent Cosecant Function Domain: x R, x nπ Range: (, 1] [1, ) x-intercept: none y-intercept: 1 Continuity: inf. discont. at x = nπ Asymptotes: x = nπ Symmetry: origin (odd function) End behavior: does not exist Graph Other Trig Functions 14 / 22

18 Example 5 Locate the vertical asymptotes and sketch the graph of y = sec 2x Graph Other Trig Functions 15 / 22

19 Example 6 Locate the ( vertical asymptotes and sketch the graph of y = csc x + π ) 3 Graph Other Trig Functions 16 / 22

20 Damped Trigonometric Functions Damped oscillation results when a sinusoid is multiplied by a function f (x) so the amplitude of the sinusoid is reduced as x approaches ± or as x approaches 0 from both directions. Graph Other Trig Functions 17 / 22

21 Example 7 Identify the damping factor f (x). Then sketch a graph of the function, f (x) and f (x). Include the viewing window from the calculator. y = x 2 sin x Graph Other Trig Functions 18 / 22

22 Example 8 Identify the damping factor f (x). Then sketch a graph of the function, f (x) and f (x). Include the viewing window from the calculator. y = x 2 cos 3x Graph Other Trig Functions 19 / 22

23 Damped Harmonic Motion When the amplitude of the motion of an object decreases with time due to friction, the motion is called damped harmonic motion. y = ke ct sin ωt y = ke ct cos ωt where c > 0 and is the damping constant, k is the displacement, t is time, and ω is the period. Graph Other Trig Functions 20 / 22

24 Example 9 A guitar string is plucked at a distance of 0.95 centimeters above its rest position, then released, causing a vibration. The damping constant for the string is 1.3, and the note produced has a frequency of 200 cycles per second. a) Write a trig function that models the motion of the string. b) Determine the amount of time t that it takes the string to be damped so that 0.38 y Graph Other Trig Functions 21 / 22

25 What You Learned You can now: Graph the parent tangent, cotangent, secant, and cosecant funtions. Graph transformations of these functions. Graph damped trigonometric functions. Do problems Chap 4.5 #1-15 odd, 17, 23, 27, odd Graph Other Trig Functions 22 / 22

26 What You Learned You can now: Graph the parent tangent, cotangent, secant, and cosecant funtions. Graph transformations of these functions. Graph damped trigonometric functions. Do problems Chap 4.5 #1-15 odd, 17, 23, 27, odd Graph Other Trig Functions 22 / 22

27 What You Learned You can now: Graph the parent tangent, cotangent, secant, and cosecant funtions. Graph transformations of these functions. Graph damped trigonometric functions. Do problems Chap 4.5 #1-15 odd, 17, 23, 27, odd Graph Other Trig Functions 22 / 22

28 What You Learned You can now: Graph the parent tangent, cotangent, secant, and cosecant funtions. Graph transformations of these functions. Graph damped trigonometric functions. Do problems Chap 4.5 #1-15 odd, 17, 23, 27, odd Graph Other Trig Functions 22 / 22

29 What You Learned You can now: Graph the parent tangent, cotangent, secant, and cosecant funtions. Graph transformations of these functions. Graph damped trigonometric functions. Do problems Chap 4.5 #1-15 odd, 17, 23, 27, odd Graph Other Trig Functions 22 / 22

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