Name:



Name: ______________________________ Date: ____________ Hr: ______

Trig/PreCalculus Trigonometry Graphs Review

1. Define each of the following in your own words:

(a) radian: (Also talk about how many radians it takes to get all the way around a circle)

(b) angular velocity:

(c) linear velocity:

(d) periodic function:

2. Change to from degrees into radians. Leave in terms of π:

(a) 225° (b) 75°

3. Change from radians to degrees:

(a) π (b) 11π

2 3

4. Find the length of the arc cut off by the 30° angle in the circle shown at left. Recall that the circumference of a circle is given by C = 2πr. Decide what fraction of a circle you have pictured to get the answer.

5. A pulley with diameter 2.0 meters is being used to lift a box. How far will the box rise if the pulley is rotated through an angle of 7π/4 radians. [Hint: Recall that the circumference all of the way around a circle is C = 2πr.]

6. Find the area of the shaded part of the circle shown at right. Recall that the area of a circle is given by A = πr2. Decide what fraction of a circle you have pictured to get the answer.

Questions 7 and 8 involve this situation: A Ferris Wheel at the county fair makes one revolution every 75 seconds. A girl rides on the Ferris Wheel, which has a radius of 20 feet.

7. Approximately what is the angular velocity of the girl in radians per second?

Recall that angular velocity = total angle

total time

8. Approximately what is the linear velocity of a girl in feet per second?

Recall that linear velocity = total distance

total time

9. Use your graphing calculator to help sketch each of the following functions. Use the ZTrig WINDOW. Finish labeling the axes.

(a) y = sin(x) (b) y = cos(x) (c) y = tan(x)

[pic] [pic] [pic]

(d) y = csc(x) (e) y = sec(x) (f) y = cot(x)

[pic] [pic] [pic]

10. Use the graphs from question 9 to answer the following questions:

(a) Suppose that n is any integer (that is, n = -2, -1, 0, 1, 2…), what is the value of:

(i) sin(nπ)? (ii) tan(nπ)? (iii) sec(nπ)?

(b) Suppose that n is any odd integer (that is, n = -3, -1, 1, 3…), what is the value of:

(i) cos(½ nπ)? (ii) csc(½ nπ)? (iii) tan(½ nπ)?

11. Draw a picture of ANY periodic graph, and use it to label the following quantities. Also write a brief definition of each.

(a) period:

(b) amplitude:

(c) vertical shift:

(d) Define frequency (may not be easy to show on your picture):

12. For each of the graphs pictured, write the amplitude and the period of the graph. Decide if it is better modeled as a sine or a cosine equation. Then write a possible equation to fit the graph:

(a) (b)

[pic] [pic]

Amplitude: __________ Amplitude: ____________

Period: _____________ Period: _______________

Sine or Cosine?_______ Sine or Cosine?________

Equation: Equation:

_____________________ _____________________

13. Use the graph shown at right below to answer the next several questions.

(a) What is the AMPLITUDE of the graph shown?

(b) What is the PERIOD of the graph shown?

(c) What is the FREQUENCY of the graph shown?

(d) What is the VERTICAL SHIFT of the graph shown?

(e) Write an equation which will match the graph shown.

14. Suppose you are given the equation y = -4cos(2π x) + 6.

7

(a) What is the AMPLITUDE of this equation?

(b) What is the PERIOD of this equation?

(c) What is the VERTICAL SHIFT of this equation?

(d) Sketch a graph of this equation. Label the axes with appropriate numbers.

15. What are the amplitude and period of the function y = 3sin( ½ x)?

16. A bungee jumper leaves the platform and then bounces up and down between position A and position C as shown in the diagram. It takes 8 seconds after she jumps for her to return to position A.

(a) Sketch a graph of the height vs. time of the bungee jumper as she bounces. Label axes with names, numbers, and units. Notice the jumper STARTS her trip (time 0) from position A.

(b) Find the following. Put units on your answer:

Period: ________________

Frequency: ________________

Amplitude: _________________

Vertical Shift: ________________

(c) Would this graph be modeled better with a sine or with a cosine formula? Write a possible formula.

17. In nature, the population of two animals, one of which preys on the other (such as foxes and rabbits) are observed to go up and down, and are found to form a sinusoidal function. The population of foxes is given by the graph below:

(a) What is the value for the amplitude? Explain how you get it.

(b) What is the value of the period? Explain how you get it.

(c) Give a formula which will calculate the number of foxes at any time.

(d) If this relationship were altered to have a smaller period, explain what that would mean in terms of foxes and time.

(e) If this relationship were altered to have a smaller amplitude, explain what that would mean in terms of foxes and time.

18. The following table gives the average monthly temperatures for the city of Phoenix, AZ.

Jan |Feb |Mar |Apr |May |June |July |Aug |Sept |Oct |Nov |Dec | |73° |80° |90° |100° |107° |110° |107° |100° |90° |80° |73° |70° | |

(a) Graph the data, putting average temperature on the y-axis and month number on the x-axis. Connect the graph up with a smooth curve. (Let t = 1 be Jan, t = 2 be Feb, etc).

(b) Find the amplitude of your graph. Put units on your answer.

(c) Find the period of your graph. Put units on your answer.

(d) Is your graph shifted up or down compared to a normal sine or cosine graph? How is it shifted and by how many units?

(e) Would the temperature best be modeled by a sine or a cosine formula? Write a possible formula for the graph.

(f) According to your model equation, what is the average temperature in August? How does this compare to the actual average?

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