Opp. Name: Date: Period: 8B.: Graphs of Cosecant and Secant Or final two trigonometric functions to graph are cosecant and secant. Remember that So, we predict that there is a close relationship between the graph of the sine and cosecant functions, and between the cosine and secant functions. Adj. Hp. Flipping Sine Waves Let s consider the relationship between values of the sine and cosecant functions. Consider this: a) Which values of give us sin = csc? b) Is csc defined for all values of? If not, for which values of undefined? What is the value of sin at these values? A P(,) 1 D c) Eplain wh the sign (positive or negative) of sin is the same as the sign of csc. (Note: this is often written as sgn(sin ) = sgn(csc ) ) d) Complete the table for some ke values of sine and cosecant sin csc 0 6 5 6 7 6 11 6 1 P a g e R e v i s e d : / 0 / 0 1 7
Eploration: Graph b hand Now we can use the values in the table above to give us a good picture of the graph of the cosecant function. 1. Plot the graph of = sin using a dotted curve.. Now draw in the vertical asmptotes for the graph of = csc (where cosecant is undefined).. Now plot each ordered pair (, csc ) on the graph below for the angles on the interval (0,) and fill in the curve for = csc 4. Finall, use the periodicit of the cosecant function to draw the graph of = csc on the interval (, 0) Some of the Same for Secant Now let s consider the relationship between the cosine and secant functions in the same wa. Consider this: a) Which values of give us cos = sec? b) Is sec defined for all values of? If not, for which values of undefined? What is the value of cos at these values? A P(,) 1 D c) Eplain wh the sign (positive or negative) of sin is the same as the sign of sec. (Note: this is often written as sgn(cos ) = sgn(sec ) ) P a g e R e v i s e d : / 0 / 0 1 7
d) Complete the table for some ke values of sine and cosecant cos sec 0 4 5 Eploration: Graph b hand Let s begin b eploring the values of the tangent function in the unit circle like we did with the sine and cosine function. Use the equations above to help. 5. Plot the graph of = cos using a dotted curve. 6. Now draw in the vertical asmptotes for the graph of = sec (where cosecant is undefined). 7. Now plot each ordered pair (, sec ) on the graph below for the angles on the interval (0,) and fill in the curve for = sec 8. Finall, use the periodicit of the cosecant function to draw the graph of = sec on the interval (, 0) P a g e R e v i s e d : / 0 / 0 1 7
Analzing the Graphs Use our graphs to find the following for f() = csc (), and = sec (Describe periodic values as an integer n ) a. Domain: = csc = sec b. Range: c. Asmptotes d. Zeros e. Maimum or Minimum? f. Smmetr (odd or even) g. Period (horizontal distance required to repeat the curve): Transforming the Cosecant and Secant Curves We now want to transform the graphs of = csc and = sec. Use our graphing calculator to eplore and determine how the constants a, b, h, and k affect the graph of f() = a csc(b( h)) + k and g() = a sec(b( h)) + k Transformations of Cosecant and Secant curves For the functions f() = a csc(b( h)) + k and g() = a sec(b( h)) + k, the constants a, b, h, and k do the following: a: Vertical stretch. The minimums and maimums are at = a and = a when k = 0. b: Horizontal Stretch. Period= b. h: Horizontal shift. k: Vertical shift. Ke to Graphing Cosecant and Secant: - To graph f() = a csc(b( h)) + k, find the zeros, maimums, & minimums of = a sin(b( h)) + k - To graph g() = a sec(b( h)) + k, find the zeros, maimums, & minimums of = a cos(b( h)) + k 4 P a g e R e v i s e d : / 0 / 0 1 7
Tr it! Determine the vertical stretch and period, and then use transformations to graph the following: Graph them without our calculator, then check them on our calculator = csc 1 1 = csc 1 5 P a g e R e v i s e d : / 0 / 0 1 7
Name: Date: Period: Assignment 8B.: Cosecant and Secant Graphs Describe how the graph the following curves differs from = csc and = sec and state the values of the asmptotes. 1. = 5 csc(). = sec ( 4 ) +. = csc ( ( 4 )) + Graph the following functions b first graphing their corresponding reciprocal function. 4. = csc 5. = 1 csc ( + ) 4 6. = sec() 7. = sec ( ) + 1 8. The U shapes in a secant or cosecant graph appear to be parabolas. Eplain wh the repeating shapes in the secant and cosecant graphs are not parabolas. 6 P a g e R e v i s e d : / 0 / 0 1 7