Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 Precalculus 1/15/2015 DO NOW/Bellwork: Do problems 1-10 on pg. 400 in the quick review section. AGENDA – HW Questions – Graphs of the other 4 trig functions Essential Question: How can the study of trigonometry be used in the real world? HOMEWORK: Pg ,17-18,21- 28,41,47-50 Objectives: You will be able to generate graphs for the tangent, cotangent, secant, and cosecant functions.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Summarizing … Standard form of the equations: y = a cos (b(x – c)) + d y = a sin (b(x – c)) + d “a” - |a| is called the amplitude, like our other functions it is like a stretch it affects “y” or the output “d” – vertical shift, it affects “y” or the output “c” – horizontal shift (phase shift), it affects “x” or “θ” or the input (don’t forget to factor out the b) “b” – period change “squishes” or “stretches out” the graph horizontal. If b > 1, the graph of the function is shrunk horizontally by a factor of 1/b. If 0 < b < 1, the graph of the function is stretched horizontally by a factor or 1/b. If b<0 then also reflection across the y-axis If |a| > 1, the amplitude stretches the graph vertically. If 0 < |a| < 1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Notice we didn’t write it as 6sin(10x- 2) that would represent a phase shift of 2/10 not 2.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Example Combining a Phase Shift with a Period Change Slide

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example Combining a Phase Shift with a Period Change Slide

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Constructing a Sinusoidal Model using Time Slide

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Constructing a Sinusoidal Model using Time Slide

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Real World Problem On a particular day in Marathon, FL high tide occurred at 9:36am. At that time the water at the end of the 33 rd Street boat ramp was 2.7 meters deep. Low tide occurred at 3:48 pm at which time the tide was only 2.1 meters deep. Assume the depth of the water is a sinusoidal function of time with a period of 12hours and 24 minutes. a)At what time on that day did the first low tide occur? b)What was the approximate depth of the water at 6am and 3pm that day? c)What was the 1 st time on that day when the water was 2.4 meters deep?

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9

10

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 The Graph of y = tan x Period: The tangent function is an odd function. The tangent function is undefined at

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 The Tangent Curve: The Graph of y = tan x and Its Characteristics

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 The Tangent Curve: The Graph of y = tan x and Its Characteristics (continued)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 The Cotangent Curve: The Graph of y = cot x and Its Characteristics

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 The Cotangent Curve: The Graph of y = cot x and Its Characteristics (continued)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 The Graphs of y = csc x and y = sec x We obtain the graphs of the cosecant and the secant curves by using the reciprocal identities We obtain the graph of y = csc x by taking reciprocals of the y-values in the graph of y = sin x. Vertical asymptotes of y = csc x occur at the x-intercepts of y = sin x. We obtain the graph of y = sec x by taking reciprocals of the y-values in the graph of y = cos x. Vertical asymptotes of y = sec x occur at the x-intercepts of y = cos x.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 The Cosecant Curve: The Graph of y = csc x and Its Characteristics

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 The Cosecant Curve: The Graph of y = csc x and Its Characteristics (continued)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 The Secant Curve: The Graph of y = sec x and Its Characteristics

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 The Secant Curve: The Graph of y = sec x and Its Characteristics (continued)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Using a Sine Curve to Obtain a Cosecant Curve Use the graph of to obtain the graph of The x-intercepts of the sine graph correspond to the vertical asymptotes of the cosecant graph.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Example: Using a Sine Curve to Obtain a Cosecant Curve (continued) Use the graph of to obtain the graph of Using the asymptotes as guides, we sketch the graph of

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 The Six Curves of Trigonometry

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 The Six Curves of Trigonometry (continued)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 The Six Curves of Trigonometry (continued)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27 The Six Curves of Trigonometry (continued)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 28 The Six Curves of Trigonometry (continued)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 29 The Six Curves of Trigonometry (continued)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 30 Find the value of x between π and 3π/2 that solves secx=-2