This document discusses the trigonometric functions csc(x) and sec(x). It provides properties of each function, including their period, domain, range, and any asymptotes or x-intercepts. Examples of graphs are shown for csc(x) with phase shifts of pi/2 and -pi/2. Properties of sec(x) are also described. The document concludes by assigning a worksheet on csc(x) and sec(x) and posing a problem to write an equation for a secant function with a given period, phase shift, and vertical shift.

2. What do we need to do?What do we need to do?

3. What do we need toWhat do we need to
change?change?

4. Graphing:Graphing:
Secant and CosecantSecant and Cosecant
Section 4.5Section 4.5

5. csc (x)csc (x)
 Flip of sin (x)Flip of sin (x)
 PropertiesProperties
– Period: 2Period: 2
– Domain: all real numbers except nDomain: all real numbers except n
– Range: (-Range: (-∞, -1] [1, ∞)∞, -1] [1, ∞)
– No x-intNo x-int
– No y-intNo y-int
– y=1 wheny=1 when
– y=-1 wheny=-1 when
π
π
nx π
π
2
2
3
+=
nx ππ 2
2
3
+=

6. Graph of csc(x)Graph of csc(x)

7. GraphGraph 2)
42
csc( ++=
πθ
y

8. GraphGraph 3)2csc( −+= πθy

9. sec (x)sec (x)
 Flip of cos (x)Flip of cos (x)
 PropertiesProperties
– Period: 2Period: 2
– Domain: all real numbers except n, n is oddDomain: all real numbers except n, n is odd
– Range: (-Range: (-∞, -1] [1, ∞)∞, -1] [1, ∞)
– No x-intNo x-int
– y-int = 1y-int = 1
– Asymptotes when , n is oddAsymptotes when , n is odd
– y=1 when , n is eveny=1 when , n is even
– y=-1 when , n is oddy=-1 when , n is odd
π
2
π
nx π=
nx π=
2
n
x
π
=

10. Graph of sec (x)Graph of sec (x)

11. GraphGraph 2)
63
sec( −+=
πθ
y

12. One last problem…One last problem…
 Write an equation for a secant functionWrite an equation for a secant function
with period , phase shiftwith period , phase shift
vertical shift -3.vertical shift -3.
π 3
π

13. HomeworkHomework
 4.5 Worksheet (csc and sec)4.5 Worksheet (csc and sec)