1. Barnett/Ziegler/Byleen Precalculus: Functions & Graphs, 4th Edition
Chapter Five
Trigonometric Functions
Copyright © 1999 by the McGraw-Hill Companies, Inc.

2. Wrapping Function
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3. Circular Functions If x is a real number and (a, b) are the
coordinates of the circular
point W(x), then:
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4. Angles (a)  positive (b)  negative (c)  and  coterminal
(a)  is a quadrantal (b)  is a third-quadrant (c)  is a second-quadrant angle angle angle

5. Angles (a) Straight angle (b) Right angle (c) Acute angle
(d) Obtuse angle

6. Radian Measure
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7. Trigonometric Functions with
Angle Domains
If q is an angle with radian measure x, then the value of each trigonometric
function at q is given by its value at the real number x.
Trigonometric
Circular
Function
Function
sin q =
sin x
cos q =
cos x
tan q
= tan x
csc q =
csc x
sec q =
sec x
cot q
= cot x
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8. Trigonometric Functions with Angle Domains Alternate Form
If q is an arbitrary angle in standard
position in a rectangular coordinate
system and P(a, b) is a point r units
from the origin on the terminal
side of q, then:
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9. Reference Triangle and Reference Angle
1. To form a reference triangle for  , draw a perpendicular from a point P(a, b) on the terminal side of  to the horizontal axis.
2. The reference angle is the acute angle (always taken positive) between the terminal side of  and the horizontal axis.
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10. 30—60  and 45  Special Triangles( 
/6) 45 2 ( 
/4) 2 1 3 45 ( 
/4) 1 60  (
/3) 1
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11. Trigonometric Functions with Angle Domains Alternate Form
If q is an arbitrary angle in standard
position in a rectangular coordinate
system and P(a, b) is a point r units
from the origin on the terminal
side of q, then:
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12. Graph of y = sin x Period: 2 Domain: All real numbers Range: [–1, 1]/2
Graph of y = sin x a b b P
(cos x
, sin x )
(0, 1) 1 x b
Period: 2
Domain: All real numbers
Range: [–1, 1]
Symmetric with respect to the origin a  2 
(–1, 0) a
(1, 0)
y = sin x = b
(0, –1) 3  /2 y 1 x       –2 – 2 3 4 -1
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13. Graph of y = cos x Period: 2 Domain: All real numbers Range: [–1, 1]/2 a b b
Graph of y = cos x P
(cos x
, sin x )
(0, 1) 1 x b
Period: 2
Domain: All real numbers
Range: [–1, 1]
Symmetric with respect to the
y axis a  2 
(–1, 0) a
(1, 0)
y = cos x = a
(0, –1) 3  /2 y 1 x –2  –   2  3  4  -1
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14. Graph of y = tan x Period:  Domain: All real numbers
except  /2 + k ,
k an integer
Range: All real numbers
Symmetric with respect to
the origin
Increasing function
between asymptotes
Discontinuous at
x =  /2 + k , k an integer 1 –2  –   2  x 5  3    3  5  – – – 2 2 2 2 2 2 –1
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15. Graph of y = cot x Period:  Domain: All real numbers except k ,
k an integer
Range: All real numbers
Symmetric with respect to
the origin
Decreasing function
between asymptotes
Discontinuous at
x = k , k an integer 1 3    3  – – 2 2 2 2 x –2  –   2  –1
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16. Graph of y = csc x y y = csc x sin 1 = y = sin x 1 x –2  –   2  –1 2  –1
Period: 2
Domain: All real numbers except k , k an integer Range: All real numbers y such that y  –1 or y  1
Symmetric with respect to the origin Discontinuous at x = k , k an integer
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17. Graph of y = sec x     y y = sec x cos 1 = y = cos x 1 x –2 – 2 –1 2  –1
Period: 2
Domain: All real numbers except /2 + k, k an integer Symmetric with respect to the y axis
Discontinuous at x = /2 + k, k an integer Range: All real numbers y such that y  –1 or y  1
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18.     Step 1. Find the amplitude | A |. Step 2. Solve Bx + C = 0 and
= 2  : Bx + C =
0 and Bx + C
= 2  C C 2 x
= – x
= – + B B B
Phase shift
Period C
2 
Phase shift = –
Period = B B
The graph completes one full cycle as Bx + C
varies from 0 to 2  —
that is, as x
varies over the interval é C C 2  ù ê – , – + ú B B B ë û é C C 2 ù
Step 3.
Graph one cycle over the interval ê – , – + ú . B B B ë û
Step 4.
Extend the graph in step 3 to the left or right as
desired.
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19. Graph of y = tan x Period:  Domain: All real numbers
except  /2 + k ,
k an integer
Range: All real numbers
Symmetric with respect to
the origin
Increasing function
between asymptotes
Discontinuous at
x =  /2 + k , k an integer 1 –2  –   2  x 5  3    3  5  – – – 2 2 2 2 2 2 –1
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20. Graph of y = cot x Period:  Domain: All real numbers except k ,
k an integer
Range: All real numbers
Symmetric with respect to
the origin
Decreasing function
between asymptotes
Discontinuous at
x = k , k an integer 1 3    3  – – 2 2 2 2 x –2  –   2  –1
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21. Graph of y = csc x y y = csc x sin 1 = y = sin x 1 x –2  –   2  –1 2  –1
Period: 2
Domain: All real numbers except k , k an integer Range: All real numbers y such that y  –1 or y  1
Symmetric with respect to the origin Discontinuous at x = k , k an integer
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22. Graph of y = sec x     y y = sec x cos 1 = y = cos x 1 x –2 – 2 –1 2  –1
Period: 2
Domain: All real numbers except /2 + k, k an integer Symmetric with respect to the y axis
Discontinuous at x = /2 + k, k an integer Range: All real numbers y such that y  –1 or y  1
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23. Facts about Inverse Functions
For f a one-to-one function and f–1 its inverse:
1. If (a, b) is an element of f, then (b, a) is an element of f–1, and conversely.
2. Range of f = Domain of f– Domain of f = Range of f–1 3.
4. If x = f–1(y), then y = f(x) for y in the
domain of f–1 and x in the domain
of f, and conversely.
5. f[f–1(y)] = y for y in the domain of f– f–1[f(x)] = x for x in the domain of f
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24. Inverse Sine Function Sine function     y y = sin x y = arcsin x æ–  2 1 x  2 –1
Sine function y –1 y
= sin x y
= arcsin x æ  ö y
= sin x  1 ,  è 2 ø æ  2 ö –  2 1
, 1 è 2 ø
(0,0)
(0,0) x x –1 1  2 æ  ö –
, –1 –1 æ  ö  è 2 ø –1
, – – è 2 ø 2 é   ù D
OMAIN = ê – , ú D
OMAIN
= [–1, 1] ë 2 2 û é   ù R
ANGE
= [–1, 1] R
ANGE = ê – , ú ë 2 2 û
Restricted sine function
Inverse sine function
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25. Inverse Cosine Functiony 1 x  –1
Cosine function y y
= cos x
= arccos –1 y y
= cos x
(–1,  ) 
(0,1) 1 è æ ø ö  2 ,0 è æ ø ö ,  2  2 x   2 –1
(1,0) ( 
, –1) x –1 1 D
OMAIN
= [0,  ] D
OMAIN
= [–1, 1] R
ANGE
= [–1, 1] R
ANGE
= [0,  ]
Restricted cosine function
Inverse cosine function
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26. Inverse Tangent Functiony y
= tan x
Tangent function 1  2 3  2 x 3  2 – –  2 –1 y y y
= tan –1 x  æ  ö y
= tan x
= arctan x 1 ,  è 4 ø 2 –  2 æ  ö 1
, 1 è 4 ø –1 x x  2 æ  1 ö –
, –1 –1 è 4 ø  2 æ  ö –1
, – – è 4 ø æ   ö D
OMAIN
= (–
 ,  ) D
OMAIN = ç – , ÷ è 2 2 ø æ   ö R
ANGE = ç – , ÷ R
ANGE
= (–  ,  ) è 2 2 ø
Restricted tangent function
Inverse tangent function
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