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The Other Trigonometric Functions Trigonometry MATH 103 S. Rook

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Overview Section 4.4 in the textbook: – Properties of the tangent & cotangent graphs – Properties of the secant & cosecant graphs 2

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Properties of the Tangent & Cotangent Graphs

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Tangent & Cotangent and the Value A Given y = A tan x or y = A cot x: – A does NOT represent amplitude y = tan x or y = cot x do not have both a minimum AND a maximum value – Recall that A affects ONLY the y-coordinates: If A > 1, the graph will be – Stretched in comparison to y = tan x or y = cot x If 0 < A < 1, the graph will be – Compressed in comparison to y = tan x or y = cot x If A < 0, the graph will be – Reflected over the x-axis 4

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Tangent & Cotangent and Period Given y = tan Bx or y = cot Bx: – Recall from Section 4.1 that the period for both y = tan x and y = cot x is π – Then y = tan Bx or y = cot Bx makes B cycles in the interval 0 to π – Thus, the period (or length of one cycle) of y = tan Bx or y = cot Bx is π ⁄ B By the interval method: 5

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Tangent & Cotangent and Vertical Translation Given y = k + tan x or y = k + cot x: – Recall that k is the vertical translation – If k > 0 y = k + tan x or y = k + cot x will be shifted UP k units as compared to y = tan x or y = cot x – If k < 0 y = k + tan x or y = k + cot x will be shifted DOWN k units as compared to y = tan x or y = cot x – The value of k affects ONLY the y-coordinate 6

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Tangent & Cotangent and Phase Shift Given y = tan(Bx + C) or y = cot(Bx + C): – The phase shift can be obtained using the interval method: Thus, the phase shift for y = tan(Bx + C) or y = cot(Bx + C) is - C ⁄ B 7

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Graphing y = k + A tan(Bx + C) or y = k + A cot(Bx + C) To graph y = k + A tan(Bx + C) or y = k + A cot(Bx + C): – Find the values for A, period, k (vertical translation), and phase shift – “Construct the Frame” for one cycle: Calculate the subinterval length (easiest to use period ⁄ 4 ) Label the x-axis by either the interval method or the formulas as previously discussed y-axis: – Make the minimum value slightly less than k + -|A| – Make the maximum value slightly more than k + |A| 8

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Graphing y = k + A tan(Bx + C) or y = k + A cot(Bx + C) (Continued) Create a table of values for the points marked on the x-axis – The tangent and cotangent will have points on the graphs that are undefined Connect the points by using the shape of the tangent or cotangent function – Recall that the tangent or cotangent will have a vertical asymptote when undefined – Extend the graph as necessary 9

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Properties of the Tangent & Cotangent Graphs (Example) Ex 1: a) identify the period b) identify the vertical translation c) identify the phase shift d) graph one cycle i) ii) 10

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Properties of the Secant & Cosecant Graphs

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Given y = k + A sec(Bx + C) or y = k + A csc(Bx + C), the properties will be the same as y = k + A tan(Bx + C) or y = k + A cot(Bx + C) EXCEPT: – Recall from Section 4.1 that the period for both y = sec x and y = csc x is 2π – Then y = sec Bx or y = csc Bx makes B cycles in the interval 0 to 2π – Thus, the period (or length of one cycle) of y = sec Bx or y = csc Bx is 2π ⁄ B By the interval method: 12

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Graphing y = k + A sec(Bx + C) or y = k + A csc(Bx + C) To graph y = k + A sec(Bx + C) or y = k + A csc(Bx + C): – Find the values for A, period, k (vertical translation), and phase shift – “Construct the Frame” for one cycle: Follow the same steps for y = k + A tan(Bx + C) or y = k + A cot(Bx + C) Be aware of undefined points on the graphs of the secant and cosecant – Vertical asymptotes will appear on the graph at these points Recall the shape of the secant and cosecant graphs – Extend the graph if necessary 13

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Properties of the Secant & Cosecant Graphs (Example) Ex 2: a) identify the period b) identify the vertical translation c) identify the phase shift d) graph on the given interval i) ii) 14

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A Final Note on Graphing Trigonometric Functions Graphing the trigonometric functions is one of the more complicated topics in the course You MUST PRACTICE to become proficient! 15

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Summary After studying these slides, you should be able to: – Identify the vertical translation, amplitude, period, and phase shift for ANY tangent, cotangent, secant, or cosecant graph or equation – Graph an equation of the form y = k + A tan(Bx + C) or y = k + A cot(Bx + C) – Graph an equation of the form y = k + A sec(Bx + C) or y = k + A csc(Bx + C) Additional Practice – See the list of suggested problems for 4.4 Next lesson – Finding an Equation from Its Graph (Section 4.5) 16

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