Here are the solutions: 1. csc = 2 2. cos = -0.4 3. sin  = 1/3 4. sin  = 1/15 5. sec = 7 6. sec = -14/26 = -0.53846 7. csc = 5

1. Characteristics of Sine and Cosine
Sine x Cosine x
Maximum: 1 Maximum: 1
Minimum: -1 Minimum: -1
Period: 360º Period: 360º
Amplitude: 1 Amplitude: 1
Zeros: 0º, 180º, 360º Zeros: 90º, 270º
y-intercept: 0 y-intercept: 1
The function is periodic The function is periodic
*Domain: 0º - 360º see *Domain: 0º - 360º see
note note
Range:-1 to 1 Range:-1 to 1
Positive trig ratios in the Positive trig ratios in the
1st and 2nd quadrant 1st and 4th quadrant
*This is not the domain of the entire sine/cosine functions but a possible domain for one
period of each
Neither
Sine x or Cosine x
The function is not
periodic
Positive trig ratios in the
2nd and 3rd quadrant
Positive trig ratios in the
3rd and 4th quadrant
The function has
asymptotes

2. Graph of Sine and Cosine in Degrees and Radians
Name _____________
Date ______________
1a) Graph y=Sine (x) using degrees.
(x-axis is in increments of 15º, y-axis is in increments of 0.5)
y
x
Characteristics:
Max. value:__________ Min. value: ___________
y intercept: __________ x intercept (zeros): ________
1b) Graph y=Sine (x) using radians.
(x-axis is in increments of
 , y-axis is in increments of 0.5)
12
y
x
Characteristics:
Max. value:__________ Min. value: ___________
y intercept: __________ x intercept (zeros): ________

3. 2a) Graph y=Cosine x using degrees.
(x-axis is in increments of 15º, y-axis is in increments of 0.5)
y
x
Characteristics:
Max. value:__________ Min. value: ___________
y intercept: __________ x intercepts (zeros): ________
2b) Graph y=Cosine x using radians.
(x-axis is in increments of
 , y-axis is in increments of 0.5)
12
y
x
Characteristics:
Max. value:__________ Min. value: ___________
y intercept: __________ x intercepts (zeros): ________

4. Graph of Sine and Cosine in Degrees and Radians

5. Complete Frayer Model for Sine and Cosine Functions
Using Radians
Complete each Frayer Model with information on each function IN RADIANS.
Period Zeros
Sine θ
Y-intercept
Characteristics
 Maximum:
 Minimum:
 Amplitude:
Period Zeros
Cosine θ
Y-intercept
Characteristics
 Maximum:
 Minimum:
 Amplitude:

6. Completed Frayer Model for Sine and Cosine Functions
Using Radians (Answers)
Complete each Frayer Model with information on each function IN RADIANS.
Period Zeros
2π Zeros: 0, π, 2π, k 
Sine θ
Y-intercept
0
Characteristics
 Maximum: 1
 Minimum: -1
 Amplitude: 1
Period Zeros
2π  , 3 ,   k
2 2 2
Cosine θ
Y-intercept
1
Characteristics
 Maximum: 1
 Minimum: -1
 Amplitude: 1

7. Reciprocal Trigonometric Functions
Name ____________________
Match the functions on the left with their reciprocals on the right.
1. sin a.
1
cos
2. cos b.
1
cot
3. tan c.
1
tan
4. sec d.
1
csc
5. csc e.
1
sin
6. cot f.
1
sec
State restrictions on each function:
(2 x  3)( x  7)
7.
( x  4)( x  2)
x(2 x  1)
8.
(3x  2)( x  2)
( x  4)( x  4)
9.
x( x  3)( x  2)
( x  7)(2 x  5)
10.
x( x  9)(3x  4)

8. Reciprocal Trigonometric Functions (Answers)
Name ____________________
Match the functions on the left with their reciprocals on the right.
1. sin D a.
1
cos
2. cos F b.
1
cot
3. tan B c.
1
tan
4. sec A d.
1
csc
5. csc E e.
1
sin
6. cot C f.
1
sec
State restrictions on each function:
(2 x  3)( x  7)
7.
( x  4)( x  2)
x  4, 2
x(2 x  1)
8.
(3x  2)( x  2)
x  2 3 , 2
( x  4)( x  4)
9. x  0,3,2
x( x  3)( x  2)
( x  7)(2 x  5)
10.
x( x  9)(3x  4)
x  0,9,  3 4

9. Graphing Secondary Trig. Functions in Radians
Complete the table as shown:
x Sin (x) Cos (x)

3

4

6

12
0

12

6

4

3
5
12

2
7
12
2
3
3
4
5
6
11
12


10. x Sin (x) Cos (x)
13
12
7
6
5
4
4
3
17
12
3
2
19
12
5
3
7
4
11
6
23
12
2
The remaining columns of the table are for the RECIPROCAL trigonometric
functions.
csc x  sec x 
1 1
You know that and .
sin x cos x
To find the values to graph these functions, simply divide “1” by each of the values
from sin x or cos x.
4
For instance, since sin  0.8660 , csc 4 
1
 1.1547
3 3 0.8660
Label the top of the extra columns with csc (x) and sec (x), and then fill in their
corresponding values.

11. What do you notice about csc0 , csc , csc2 , sec  , sec 3 ?
2 2
Why does this happen?
What occurs on the graphs of the reciprocals at those points?
State the restrictions of the secant and cosecant functions:
Secant:
Cosecant:

12. Answers
Sine x Cosine x
Period: 2 Period: 2
Maximum Point: Maximum Points:
   0,1  2 ,1
 ,1
2 
Minimum Point:
Minimum Point:  , 1
 3 
 , 1
 2  Y-intercept: 1
Y-intercept: 0  3
Zeros: ,
2 2
Zeros: 0, 2

13. Completed table as shown:
x Sin (x) Csc (x) Cos (x) Sec (x)
 -0.8660 -1.155 0.5 2
3
 -0.7071 -1.414 0.7071 1.4142
4
 -0.5 -2 0.8660 1.1547
6
 -0.2588 -3.864 0.9659 1.0353
12
0 0 ERROR 1 1
 0.2588 3.8637 0.9659 1.0353
12
 0.5 2 0.8660 1.1547
6
 0.7071 1.4142 0.7071 1.4142
4
 0.8660 1.1547 0.5 2
3
5 0.9659 1.0353 0.2588 3.8637
12
 1 1 0 ERROR
2
7 0.9659 1.0353 -0.2588 -3.864
12
2 0.8660 1.1547 -0.5 -2
3
3 0.7071 1.4142 -0.7071 -1.414
4
5 0.5 2 -0.8660 -1.155
6
11 0.2588 3.8637 -0.9659 -1.035
12
 0 ERROR -1 -1

14. Answers continued
x Sin (x) Csc (x) Cos (x) Sec (x)
13 -0.2588 -3.864 -0.9659 -1.035
12
7 -0.5 -2 -0.8660 -1.155
6
5 -0.7071 -1.414 -0.7071 -1.414
4
4 -0.8660 -1.155 -0.5 -2
3
17 -0.9659 -1.035 -0.2588 -3.864
12
3 -1 -1 0 ERROR
2
19 -0.9659 -1.035 0.2588 3.8637
12
5 -0.8660 -1.155 0.5 2
3
7 -0.7071 -1.414 0.7071 1.4142
4
11 -0.5 -2 0.8660 1.1547
6
23 -0.2588 -3.864 0.9659 1.0353
12
2 0 ERROR 1 1
The remaining columns of the table are for the RECIPROCAL trigonometric
functions.
csc x  sec x 
1 1
You know that and .
sin x cos x
To find the values to graph these functions, simply divide “1” by each of the values
from sin x or cos x.
4
For instance, since sin  0.8660 , csc 4 
1
 1.1547
3 3 0.8660
Label the top of the extra columns with csc (x) and sec (x) , then fill in their
corresponding values.

15. Answers continued
What do you notice about csc0 , csc , csc2 , sec  , sec 3 ?
2 2
ERROR
Why does this happen?
Because you are dividing by zero, which is undefined
What occurs on the graphs of the reciprocals at those points?
Vertical lines
State the restrictions of the secant and cosecant functions:
Secant: x   , 3 nor any decrease or increase by 
2 2
Cosecant: x  0, ,2 nor any of their multiples

16. Reciprocal Trigonometric Functions Practice
Knowledge
Find each function value:
csc , cos , if sec  2.5
2
1. if sin   2.
4
3. sin  , if csc  3 4. sin  , if csc  15
1
sec , if cos  sec , if cos 
5
5. 6.
7 26
 14
csc , if sin   cos , if sec 
11
7. 8.
6 3
sin  , if csc sec , if cos
3 6
9.  10. 
3 12
Application
Find each function value (keep answers in radical form):
csc tan   sec , if sin  
6 3
11. , if 12.
12 3
 3
cos , if cot   sin  , if cos 
3
13. 14.
3 2
15. sec , if csc  15 16. cos , if csc  15
2
17. sec , if tan   3 18. csc , if sin  
12
2
cos , if sin   sin  , if tan  
5
19. 20.
13 5
ANSWERS:
4 1 1
1. 2. -0.4 3. 4. 5.  7
2 3 15
26 6 3 3 12
6. 7. 8.  9. 10.
5 11 14 3 6
6 1 1 15
11. 5 12. 13.  14. 15.
3 2 2 14
14 12 12 2
16. 17. 2 18.  19. 20. 
15 2 13 3

17. Characteristics of Tangent and Cotangent Functions
Tangent x Cotangent x
No maximum No maximum
No minimum No minimum
Period: 180º Period: 180º
Zeros: 0º, 180º, 360º Zeros: 90º, 270º
y-intercept: 0 y-intercept: 1

18. Graphs of Tangent and Cotangent in Degrees
On the given set of axes, graph Tangent x and Cotangent x.
(x-axis is in increments of 15º)
(y-axis is in increments of 0.5)
y = Tangent (x)
y
x
Characteristics:
y = Cotangent (x)
y
x
Characteristics:

19. Graphs of Tangent and Cotangent in Radians
On the given set of axes, graph Tangent x and Cotangent x.
(x-axis is in increments of
 )
12
(y-axis is in increments of 0.5)
y = Tangent (x)
y
x
Characteristics:
y = Cotangent (x)
y
x
Characteristics:

20. Graphs of Tangent and Cotangent in Radians (Answers)
In the solution given for cotx=- the graph does not have any holes, only asymptotes

21. Frayer Model for Tangent and Cotangent
Complete each Frayer Model with information on each function IN RADIANS.
Period Zeros
Tangent θ
Y-intercept
Characteristics
 Maximum:
 Minimum:
 Asymptotes:
Period Zeros
Cotangent θ
Y-intercept
Characteristics
 Maximum:
 Minimum:
 Asymptotes:

22. Frayer Model for Tangent and Cotangent (Answers)
Complete each Frayer Model with information on each function IN RADIANS.
Period Zeros
0,  , 2

Tangent θ
Y-intercept
Characteristics
0
 Maximum: None
 Minimum: None
 3
 Asymptotes: ,
2 2
Period Zeros
 None
Cotangent θ
Y-intercept
Characteristics
None
 Maximum: None
 3  Minimum: None
‘Holes’ at ,
2 2  Asymptotes: 0,  , 2

23. Rate of Change for Trigonometric Functions
Given the function:

f ( )  3sin  

 6

  
1. Sketch f   on an interval  , 76 
6
 

2. Is the function increasing or decreasing on the interval
 to
2
.
3 3
   2 
3. Draw the line through the points f   and f 
3  3 
 
4. Find the average rate of change of the function f ( )  3sin   
  from
 to
 6 3
2
.
3
5. What does this mean?
6. Describe how to find the instantaneous rate of change of

f ( )  3sin   
 at
 6
 . What does this mean?
3

24. Rate of Change for Trigonometric Functions (Answers)
Given the function:

f ( )  3sin   

 6
*And the points:
 2
3 3
  7 
1. Sketch on an interval  , 
6 6 
 
2. Is the function increasing or decreasing on the interval
 to
2
. Increasing
3 3
  2 
3. Draw the line through the points f 
  and f 
3    3 
 
4. Find the average rate of change of the function

f ( )  3sin   
 from
 to
 6 3
2
.
3
f     f  2 
   
3
  1.5  3  1.5  0.025
   3 

  2  
3 3 3 3

25. (Answers continued)
5. What does this mean?
   2 
This is the slope of the line through the points  ,1.5  and  ,3 
3   3 
6. Find the instantaneous rate of change at
 .
3
 
To find instantaneous rate of change at , choose values for θ which move closer to
3 3
2
from .
3
   
f   f 
 2  3   2.5981  1.5  1.0981  0.0366
At
2    

2 3 6 6
 5   
5
f   f   2.1213  1.5 0.6213
At  12  3   0.0414
12 5   

12 3 12 12
 7   
7
f   f   1.9284  1.5 0.4284
At  18  3   0.0428
18 7   

18 3 18 18
 13   
13
f   f   1.7207  1.5 0.2207
At  36  3   0.0441
36 13   

36 3 36 36
 61   
61
f   f   1.5451  1.5 0.0451
At  180  3   0.0451
180 61  1 1

180 3
Approaches 0.05. This means that the slope of the line tangent to
f ( )  3sin     at 
 
is 0.05
 6 3
 

26. Rate of Change for Trigonometric Functions: Problems
Practice and participation Task
For each of the following functions, sketch the graph on the indicated interval. Find the
average rate of change using the identified points, and then find the instantaneous rate
of change at the indicated point.
1. In a simple arc for an alternating current circuit, the current at any instant t is
given by the function f (t) =15sin (60t). Graph the function on the interval 0 ≤ t ≤
5. Find the average rate of change as t goes from 2 to 3. Find the instantaneous
rate of change at t = 2.
2. The weight at the end of a spring is observed to be undergoing simple harmonic
motion which can be modeled by the function D (t) =12sin (60π t). Graph the
function on the interval 0 ≤ t ≤ 1. Find the average rate of change as t goes from
0.05 to 0.40. Find the instantaneous rate of change at t = 0.40.
3. In a predator-prey system, the number of predators and the number of prey tend
to vary in a periodic manner. In a certain region with cats as predators and mice
as prey, the mice population M varied according to the equation
M=110250sin(1/2)π t, where t is the time in years since January 1996. Graph the
function on the interval 0≤ t ≤ 2. Find the average rate of change as t goes from
0.75 to 0.85. Find the instantaneous rate of change at t = 0.85.
4. A Ferris Wheel with a diameter of 50 ft rotates every 30 seconds. The vertical
position of a person on the Ferris Wheel, above and below an imaginary
horizontal plane through the center of the wheel can be modeled by the equation
h (t) =25sin12t. Graph the function on the interval 15 ≤ t ≤ 30. Find the average
rate of change as t goes from 24 to 24.5. Find the instantaneous rate of change
at t = 24.
5. The depth of water at the end of a pier in Vacation Village varies with the tides
throughout the day and can be modeled by the equation
D=1.5cos [0.575(t-3.5)] + 3.8. Graph the function on the interval 0 ≤ t ≤ 10. Find
the average rate of change as t goes from 4.0 to 6.5. Find the instantaneous rate
of change at t = 6.5.

27. Rate of Change for Trigonometric Functions: Problems
(Answers)
1.
AVERAGE RATE OF INSTANTANEOUS
CHANGE = -12.99 RATE OF CHANGE = -8
2.
AVERAGE RATE OF INSTANTANEOUS
CHANGE = 27.5629 RATE OF CHANGE =
10
3.
AVERAGE RATE OF INSTANTANEOUS
CHANGE = 53460 RATE OF CHANGE =
40,000
4.
AVERAGE RATE OF INSTANTANEOUS
CHANGE = 1.88 RATE OF CHANGE =
1.620
5.
AVERAGE RATE OF INSTANTANEOUS
CHANGE = -0.66756 RATE OF CHANGE = -
0.9