1 / 16

The Other Trigonometric Functions

The Other Trigonometric Functions. Trigonometry MATH 103 S. Rook. Overview. Section 4.4 in the textbook: Properties of the tangent & cotangent graphs Properties of the secant & cosecant graphs. Properties of the Tangent & Cotangent Graphs. Tangent & Cotangent and the Value A.

nerita
Download Presentation

The Other Trigonometric Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

  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

  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:

  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 UPk 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 DOWNk units as compared to y = tan x or y = cot x • The value of kaffects ONLY the y-coordinate

  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

  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|

  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

  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)

  11. Properties of the Secant & Cosecant Graphs

  12. Properties of the Secant & Cosecant Graphs • 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:

  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

  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)

  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!

  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)

More Related