1. The Other Trigonometric Functions Trigonometry MATH 103 S. Rook

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

3. Properties of the Tangent & Cotangent Graphs

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. Properties of the Secant & Cosecant Graphs

12. 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

13. 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

14. 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

15. 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

16. 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