Pre Calculus Graphing Trig. Functions Day 1 Draw a graph of the given trigonometric function with the listed amplitude and period. 1. cosy x 2. siny x

Amp: ½ Period: 4

Amp: 5 Period: 10π

Draw a graph of the given trigonometric function with the listed vertical scale change and period. 3. tany x 4. tany x 5. tany x

v. stretch: 3 period: π v. shrink: 14 period: 6 π v. stretch: –6 period:

5

6. For each tangent function graphed above in questions 3-5, a) List the equations of two vertical asymptotes of each graph.

b) What relationship do you see between the location of the vertical asymptotes and the period of tangent? Do you think this will always be true?

For each function below, identify the transformation from the parent trigonometric function. Then, state the amplitude and period.

7. y = 3 cos (2x) 8. y = – sin (3x) 9. 4 tan2

xy

State the (a) amplitude (if applicable) and (b) period. For tangent, list the vertical stretch or shrink. 10. 1

54cosy x 11. 5sin 4y x 12. 3sin 7y x 13. cosy x

14. 4

tanx

y

15. tan 3y x 16. tan 23y x

17. Graph at least one period for EACH of the functions in questions 10-16. Attach a separate piece of paper and be sure to label ALL tick marks on your axes. 18. The alternating current I, in amperes, flowing through an ac (alternating current) circuit at time t is

( )120sin 30I tp= , where t > 0.

a) What is the period? b) What is the amplitude? c) Graph this function over two periods. Write two equations of the sine function with each amplitude and period.

19. amplitude = 5, period = 2 20. amplitude = 2

3 , period =

Write two equations of the cosine function with each amplitude and period.

21. amplitude = 1 , period = 2

3

22. amplitude =

1

2, period =

4

In questions 23 –26, list the amplitude (or vertical stretch) and period. Then, write the equation of the given graph. 23. 24. 25. 26.

Pre Calculus Graphing Trig. Functions Day 2

1. Consider the function 123cosy x .

a) State the amplitude, period, phase shift and vertical shift.

If the ½ is distributed inside the cosine function, we could write this function as 12 23cosy x .

In this form, the phase shift (or horizontal translation) is NOT 2p .

b) Take the inside of the cosine function above, set it equal to zero, and solve. What does the solution tell you? When the inside of a function is of the form (bx – c), there are two ways to determine the phase shift:

i) Factor out the b value … ( )( )cbb x-

ii) Set the inside of the function equal to zero and solve for x … 0 cbbx c x- = =

c) State the amplitude, period, phase shift, and vertical shift of cos 3 5y x

One concept that is particularly difficult for students is when the phase shift is not already one of the tick marks on the x axis. This happens when the phase shift is not a factor of the period. We noticed this on question 6, but we simply divided the scale on our x-axis into smaller pieces. It is not always so simple…

2. Given a period of and a phase shift of 3

, create the scale you would use for the x axis. Hint: you may want to think

of a common denominator.

3. Given a period of 2

and a phase shift of

6

, create the scale you would use for the x axis. Hint: you may want to

think of a common denominator.

0

0

State the amplitude (if applicable), period, phase shift and vertical shift of each function. Then, graph at least one period of each function ON GRAPH PAPER…I have plenty!! Be sure to label the scale on your axes.

4. cos 2 33

y x

5. tan 2 3y x 6.

1 3cos

4 8y x

7. 123siny x 8. tan 2

3y x

9. 1sin 4 2

2y x

Write two equations of the cosine function with:

10. amplitude = 1, period = 4

, phase shift = , vertical shift = –1

Write two equations of the sine function with:

11. amplitude = 6, period = 2

, phase shift =

8

, vertical shift none.

In questions 12-14, list the amplitude, period, phase shift, and vertical shift of each graph. Then, write the equation of the graph using the indicated parent function. 12. xy sin 13. xy tan 14. xy cos

p

1

x

y

p

1 x

y

1x

y

p-

For questions 15 and 16, write both a sine AND a cosine function for each function shown. 15. 16. 17. The following data represent the average monthly temperatures for Juneau, Alaska.

Month (x)

1 (Jan)

2 (Feb)

3 (Mar)

4 (Apr)

5 (May)

6 (Jun)

7 (Jul)

8 (Aug)

9 (Sep)

10 (Oct)

11 (Nov)

12 (Dec)

Average Monthly

Temperature (F) 24.2 28.4 32.7 39.7 47.0 53.0 56.0 55.0 49.4 42.2 32.0 27.1

a) Your calculator is capable of finding a Sinusoidal Regression. Find the sine function that fits the data. b) What is the amplitude? c) What is the period? d) What is the phase shift? e) What is the vertical shift?

Pre Calculus Worksheet: Review of Inverses (prerequisites before 4.7) 1. Define Inverse. 2. How do you find the inverse of an equation? The inverse of a graph? 3. Find the inverse of each function. Then, tell the Domain and Range of the Inverse.

a) ( ) 3 xg x

x

+= b) ( ) 4 2 1k x x= + - c) ( ) 27 xm x e +=

4. Determine which, if any, of the inverses in question 3 are NOT functions. Explain why not. 5. Graph the inverse of the given relation. Tell whether each inverse is a function or is not a function. Explain.

a. b. c. 2y x 6. How can you PREDICT whether a function’s inverse will be a function?

7. What did we do first semester to 2y x so that it its inverse was a function?

8. Now, let’s try some inverse trig graphs. Remember to think of the 5 key points from the Parent Functions.

a. 1sin ( )y x b. 1cos ( )y x c. 1tan ( )y x

Think of what we did to 2y x to make the inverse a function first semester... 9. Now highlight a piece of each graph that is a function. The piece you want must include the angles on the interval

2(0, ) because these angles are in a right triangle. Think about the attributes of the parent function that you should also

include. You do NOT have to use the same piece for all three graphs. List an interval for the angles you want to include.

a. 1sin ( )y x b. 1cos ( )y x c. 1tan ( )y x

10. Highlight the part of the unit circle corresponding to the angles you selected in question 9.

a. 1sin ( )y x b. 1cos ( )y x c. 1tan ( )y x

Finally, let’s think of input and outputs…

11. For the original trig. graphs ( sin , cos , tany x y x y x ), x gives an ______________ and y gives a ratio of the

sides of a right triangle. For the inverse trig. graphs ( 1 1 1sin , cos , tany x y x y x ), x gives __________________

and y gives __________________.

1

π

1

π

1

π

Pre Calculus Worksheet 4.7 1. The “input” to a trig function is an _______________, and the “output” is a ______________. 2. Evaluate each of the following trig expressions:

a) 3

cos4

b) 7

sin6

c)

3

5tan

d) sec

2

5

3. The “input” to an inverse trig function is a _____________, and the “output” is an ________________. 4. For each of the following, what quadrant is the answer in? (You don’t have to find the answer.) a) ( )1sin 0.8- b) ( )1cos 0.8- c) ( )1tan 0.8-

5. Based on your answers to question 4, what conclusion can you make when the “input” of an inverse trig function is a positive value? 6. For each of the following, what quadrant is the answer in? (You don’t have to find the answer.) a) ( )1sin 0.8- - b) ( )1cos 0.8- - c) ( )1tan 0.8- -

7. The following represent EVERY inverse sine function that you are expected to know w/o a calculator.

a) 1 1sin

2- æ ö÷ç ÷ç ÷çè ø

b) 1

arcsin2

c) 1 1sin

2

d) 1 1sin

2

e) sin–1 (–1) f) arcsin(1)

g) ( )1sin 0- h) 1 3sin

2-æ ö÷ç ÷ç ÷ç ÷÷çè ø

i) 1 3sin

2-æ ö÷ç ÷ç- ÷ç ÷÷çè ø

8. The following represent EVERY inverse cosine function that you are expected to know w/o a calculator.

a) 1 1cos

2- æ ö÷ç ÷ç ÷çè ø

b) 1

arccos2

c) 1 1cos

2

d) 1 1cos

2

e) cos–1 (–1) f) arccos(1)

g) ( )1cos 0- h) 1 3cos

2-æ ö÷ç ÷ç ÷ç ÷÷çè ø

i) 1 3cos

2-æ ö÷ç ÷ç- ÷ç ÷÷çè ø

9. The following represent EVERY inverse tangent function that you are expected to know w/o a calculator.

a) 1 1tan

3- æ ö÷ç ÷ç ÷÷çè ø

b) 3

arctan3

c) 1tan 3

d) 1tan 3 e) tan–1 (–1) f) arctan(1)

g) ( )1tan 0-

Find the exact value without a calculator.

10. 1sin cos4

11. 1 7cos cos

4

12. 1 1cos 2sin

2

13. 1csc tan 1

Find an algebraic expression equivalent to the given expression.

14. 1sin tan x 15. 1cos tan x

16. tan arcsin x 17. sin arccos 3x

For questions 18 and 19, use a calculator to find the approximate value in degrees.

18. arcsin 0.67 19. 1tan 2.37 Most calculators do not have keys for evaluating the inverse cotangent, cosecant, and secant functions. The easiest way to do this is to first convert the inverse expression into a normal cotangent, cosecant, or secant function, then use the reciprocal properties and their inverses to evaluate.

While ( )( )

1sec

cosx

x= , NOTICE that ( )

( )1

1

1sec

cosx

x-

-¹

For questions 20 – 22, evaluate each using your calculator. Be sure you are in radian mode.

20. ( )1sec 3- 21. 1 1cot

2- æ ö÷ç ÷ç ÷çè ø

22. ( )1csc 4- -

A B

d

14 34

30 km

Pre Calculus Worksheet 4.8 1. The length of a shadow of a tree is 130 feet when the angle of elevation to the sun is θ. a) Draw a picture and write the height of the tree as a function of θ.

b) If θ = 25, find the height of the tree.

c) Rewrite your function in part a so that θ is a function of the height of the tree. d) If the height of the tree is 100 feet, what is the angle of elevation? 2. An observer in a lighthouse 350 feet above sea level observes two ships directly offshore. The angles of depression to

the ships are 4 and 6.5. How far apart are the ships?

3. Two fire towers are 30 kilometers apart, tower A being due west of tower B. A fire is spotted from the towers, and the

bearings from A and B are E 14 N and W 34 N, respectively.

Find the distance d of the fire from the line segment AB. 4. The sonar of a navy cruiser detects a nuclear submarine that is 4000 feet from the cruiser. The angle between the water

level and the submarine is 31.5. How deep is the submarine?

5. A plane is 160 miles north and 85 miles east of an airport. If the pilot wants to fly directly to the airport, what bearing should be taken?

110

100550 ft

6. A shoreline runs north-south, and a boat is due east of the shoreline.

The bearings of the boat from two points on the shore are 110 and 100. Assume the points are 550 feet apart. How far is the boat from the shore?

7. The horizontal distance that a projectile will travel in the air is given by the equation ( ) ( )2

0 sin 2vR

g

q= , where 0v is the

initial velocity of the projectile, q is the angle of elevation, and g is acceleration due to gravity. For this problem, let’s assume g = 9.8 m/s2.

a) Suppose you are able to throw the ball with an initial velocity of 40 m/s. How much further will the ball

travel, if the angle of elevation is changed from 20 to 25 ?

b) If you can throw a baseball with an initial speed of 34.8 m/s, at what angle of elevation should you direct the throw so that the ball travels a distance of 107 meters before striking the ground?

Simple Harmonic Motion refers to motion that repeats in a regular pattern. The bouncing motion of a mass attached to a spring is a good example of simple harmonic motion. The displacement of the mass as a function of time is a sine or a cosine function. The displacement is the amplitude, and the time it takes to complete one full cycle of motion is the period. 8. A mass on a spring oscillates back and forth and completes one cycle in 0.8 seconds. Its maximum displacement is 4 cm. Write an equation that models this motion. (Assume the mass begins in the middle).

0

4 cm

– 4 cm

8 feet

25 feet

9. The Ferris wheel shown makes one complete turn every 20 seconds. A rider’s height, h, above the ground can be modeled by the equation ( )sinh a t kw= + , where h and k are given in feet and t is in seconds.

a) What is the value of a ? b) What is the value of k ? c) What is the value of w ?

d) Using this model, where is the person when they first get on the ride? Is this realistic?

e) Write a different equation for h that might be a little more realistic in terms of where a person gets on the ride. 10. The Halo: Reach Arena rating system is based on the following formula:

Rating = ( )( )120001000 tanL R Np-+ ⋅ ,

where L = loss penalty, R = raw rating, and N = game normalization.

L = 1 if you win, and L = 0.96 if you lose. In other words, losing the game means your rating decreases 4%.

R = kills + assists – deaths

3 – betrayals N =

# of players on each team

score needed to win

a) What is your initial rating? … Let R = 0 & L = 1

b) Your team consisted of 8 players, and you won when you scored 50 points. You also had 15 kills, 4 assists, 12 deaths and 1 betrayal. Find your Rating.

c) Losing is simply a rating penalty, so let’s suppose that you win, which means L = 1. Most games have either 4 players on each team with a score of 25 needed to win or 8 players with a score of 50 to win … either way, N = 0.16. Using what you know about the inverse tangent function (try reasoning this out without a calculator if possible), what is the maximum rating possible according to this formula? What about the minimum? Will you ever reach either of these ratings? [Hint … if you are stuck, graph the Rating formula in your calculator and adjust your window … USE x = R]