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 note

*Domain: 0º - 360º see note

Range:-1 to 1 Range:-1 to 1 Positive trig ratios in the 1st and 2nd quadrant

Positive trig ratios in the 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

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)

Characteristics: Max. value:__________ Min. value: ___________ y intercept: __________ x intercept (zeros): ________ 1b) Graph y=Sine (x) using radians.

(x-axis is in increments of 12

, y-axis is in increments of 0.5)

Characteristics: Max. value:__________ Min. value: ___________ y intercept: __________ x intercept (zeros): ________

x

y

x

y

2a) Graph y=Cosine x using degrees. (x-axis is in increments of 15º, y-axis is in increments of 0.5)

Characteristics: Max. value:__________ Min. value: ___________ y intercept: __________ x intercepts (zeros): ________ 2b) Graph y=Cosine x using radians.

(x-axis is in increments of 12

, y-axis is in increments of 0.5)

Characteristics: Max. value:__________ Min. value: ___________ y intercept: __________ x intercepts (zeros): ________

x

y

x

y

Graph of Sine and Cosine in Degrees and Radians

Complete Frayer Model for Sine and Cosine Functions Using Radians

Complete each Frayer Model with information on each function IN RADIANS. Period

Zeros

Y-intercept

Characteristics

Maximum: Minimum: Amplitude:

Period

Zeros

Y-intercept

Characteristics

Maximum: Minimum: Amplitude:

Sine θ

Cosine θ

Completed Frayer Model for Sine and Cosine Functions Using Radians (Answers) Complete each Frayer Model with information on each function IN RADIANS. Period 2π

Zeros Zeros: 0, π, 2π, k

Y-intercept 0

Characteristics

Maximum: 1 Minimum: -1 Amplitude: 1

Period 2π

Zeros

2

, 32

,2

k

Y-intercept 1

Characteristics

Maximum: 1 Minimum: -1 Amplitude: 1

Sine θ

Cosine θ

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:

7. (2 3)( 7)

( 4)( 2)

x x

x x

8. (2 1)

(3 2)( 2)

x x

x x

9. ( 4)( 4)

( 3)( 2)

x x

x x x

10. ( 7)(2 5)

( 9)(3 4)

x x

x x x

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:

7. (2 3)( 7)

( 4)( 2)

x x

x x

4, 2x

8. (2 1)

(3 2)( 2)

x x

x x

2 , 23x

9. ( 4)( 4)

( 3)( 2)

x x

x x x

0,3,2x

10. ( 7)(2 5)

( 9)(3 4)

x x

x x x

3

40,9,x

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

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.

You know that 1

sincsc

xx and

1

cossec

xx .

To find the values to graph these functions, simply divide “1” by each of the values from sin x or cos x.

For instance, since 4

0.86603

sin ,

4 11.1547

3 0.8660csc

Label the top of the extra columns with csc (x) and sec (x), and then fill in their corresponding values.

What do you notice about csc0 , csc , csc2 , 2

sec , 3

2sec

?

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:

Answers

Sine x

Period: 2 Maximum Point:

,12

Minimum Point:

3

, 12

Y-intercept: 0 Zeros: 0,2

Cosine x

Period: 2 Maximum Points:

0,1 2 ,1

Minimum Point:

, 1

Y-intercept: 1

Zeros: 2

,

3

2

Completed table as shown: x Sin (x) Csc (x) Cos (x) Sec (x)

3

-0.8660 -1.155 0.5 2

4

-0.7071 -1.414 0.7071 1.4142

6

-0.5 -2 0.8660 1.1547

12

-0.2588 -3.864 0.9659 1.0353

0

0 ERROR 1 1

12

0.2588 3.8637 0.9659 1.0353

6

0.5 2 0.8660 1.1547

4

0.7071 1.4142 0.7071 1.4142

3

0.8660 1.1547 0.5 2

5

12

0.9659 1.0353 0.2588 3.8637

2

1 1 0 ERROR

7

12

0.9659 1.0353 -0.2588 -3.864

2

3

0.8660 1.1547 -0.5 -2

3

4

0.7071 1.4142 -0.7071 -1.414

5

6

0.5 2 -0.8660 -1.155

11

12

0.2588 3.8637 -0.9659 -1.035

0 ERROR -1 -1

Answers continued

x Sin (x) Csc (x) Cos (x) Sec (x) 13

12

-0.2588 -3.864 -0.9659 -1.035

7

6

-0.5 -2 -0.8660 -1.155

5

4

-0.7071 -1.414 -0.7071 -1.414

4

3

-0.8660 -1.155 -0.5 -2

17

12

-0.9659 -1.035 -0.2588 -3.864

3

2

-1 -1 0 ERROR

19

12

-0.9659 -1.035 0.2588 3.8637

5

3

-0.8660 -1.155 0.5 2

7

4

-0.7071 -1.414 0.7071 1.4142

11

6

-0.5 -2 0.8660 1.1547

23

12

-0.2588 -3.864 0.9659 1.0353

2

0 ERROR 1 1

The remaining columns of the table are for the RECIPROCAL trigonometric functions.

You know that 1

sincsc

xx and

1

cossec

xx .

To find the values to graph these functions, simply divide “1” by each of the values from sin x or cos x.

For instance, since 4

0.86603

sin ,

4 11.1547

3 0.8660csc

Label the top of the extra columns with csc (x) and sec (x) , then fill in their corresponding values.

Answers continued

What do you notice about csc0 , csc , csc2 , 2

sec , 3

2sec

?

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: 3

,2 2

x nor any decrease or increase by

Cosecant: 0, ,2x nor any of their multiples

Reciprocal Trigonometric Functions Practice

Find each function value:

1. csc , if 2

sin4

2. cos , if sec 2.5

3. sin , if csc 3 4. sin , if 15csc

5. sec , if 1

7cos 6. sec , if

5

26cos

7. csc , if 11

6sin 8. cos , if

14

3sec

9. sin , if 3

3csc 10. sec , if

6

12cos

Find each function value (keep answers in radical form):

11. csc , if 6

12tan 12. sec , if

3

3sin

13. cos , if 3

3cot 14. sin , if

3

2cos

15. sec , if 15csc 16. cos , if 15csc

17. sec , if 3tan 18. csc , if 2

12sin

19. cos , if 5

13sin 20. sin , if

2

5tan

Knowledge

Application

ANSWERS:

1. 4

2 2. -0.4 3.

1

3 4.

1

15 5. 7

6. 26

5 7.

6

11 8.

3

14 9.

3

3 10.

12

6

11. 5 12. 6

3 13.

1

2 14.

1

2 15.

15

14

16. 14

15 17. 2 18.

12

2 19.

12

13 20.

2

3

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

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)

Characteristics: y = Cotangent (x)

Characteristics:

x

y

x

y

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)

Characteristics: y = Cotangent (x)

Characteristics:

x

y

x

y

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

Frayer Model for Tangent and Cotangent Complete each Frayer Model with information on each function IN RADIANS. Period

Zeros

Y-intercept

Characteristics

Maximum: Minimum: Asymptotes:

Period

Zeros

Y-intercept

Characteristics

Maximum: Minimum: Asymptotes:

Tangent θ

Cotangent θ

Frayer Model for Tangent and Cotangent (Answers) Complete each Frayer Model with information on each function IN RADIANS. Period

Zeros

0, , 2

Y-intercept

0

Characteristics

Maximum: None Minimum: None

Asymptotes: 3

,2 2

Period

Zeros None

Y-intercept None

‘Holes’ at 3

,2 2

Characteristics

Maximum: None Minimum: None

Asymptotes: 0, , 2

Tangent θ

Cotangent θ

Rate of Change for Trigonometric Functions

Given the function: ( ) 3sin6

f

1. Sketch f on an interval 7

6 6,

2. Is the function increasing or decreasing on the interval 3

to 23

.

3. Draw the line through the points 3

f

and 23

f

4. Find the average rate of change of the function ( ) 3sin6

f

from 3

to

23

.

5. What does this mean?

6. Describe how to find the instantaneous rate of change of ( ) 3sin6

f

at

3

. What does this mean?

Rate of Change for Trigonometric Functions (Answers)

Given the function: ( ) 3sin6

f

*And the points: 3

23

1. Sketch on an interval 7

6 6,

2. Is the function increasing or decreasing on the interval 3

to 23

. Increasing

3. Draw the line through the points 3

f

and 23

f

4. Find the average rate of change of the function ( ) 3sin6

f

from 3

to

23

.

23 3

23 3

f f

1.5 3

3

1.5

3

0.025

(Answers continued) 5. What does this mean?

This is the slope of the line through the points ,1.53

and 2

,33

6. Find the instantaneous rate of change at 3

.

To find instantaneous rate of change at 3

, choose values for θ which move closer to

3

from 2

3

.

At 2

2.5981 1.5 1.09812 30.0366

2 3 6 6

f f

At 5

12

52.1213 1.5 0.621312 3

0.0414512 3 12 12

f f

At 7

18

71.9284 1.5 0.428418 3

0.0428718 3 18 18

f f

At 13

36

131.7207 1.5 0.220736 3

0.04411336 3 36 36

f f

At 61

180

611.5451 1.5 0.0451180 3

0.045161 1 1180 3

f f

Approaches 0.05. This means that the slope of the line tangent to

( ) 3sin6

f

at 3

is 0.05

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.

Rate of Change for Trigonometric Functions: Problems (Answers)

1.

AVERAGE RATE OF CHANGE = -12.99

INSTANTANEOUS RATE OF CHANGE = -8

2.

AVERAGE RATE OF CHANGE = 27.5629

INSTANTANEOUS RATE OF CHANGE = 10

3.

AVERAGE RATE OF CHANGE = 53460

INSTANTANEOUS RATE OF CHANGE = 40,000

4.

AVERAGE RATE OF CHANGE = 1.88

INSTANTANEOUS RATE OF CHANGE = 1.620

5.

AVERAGE RATE OF CHANGE = -0.66756

INSTANTANEOUS RATE OF CHANGE = -0.9