Mathematics IV Frameworks

Foundations of Math III

Unit 5: Trigonometry

Academics High School Mathematics Adapted from: Georgia Department of Education

2 5.1: Getting Started with Trigonometry and the Unit Circle

An angle is in standard position when the vertex is at the origin and the initial side lies on the positive side of the x-axis.

The ray that forms the initial side of the angle is rotated around the origin with the resulting ray being called the terminal side of the angle.

An angle is positive when the location of the terminal side results from a counterclockwise rotation. An angle is negative when the location of the terminal side results from a clockwise rotation.

terminal side counterclockwise 130

initial side clockwise -230

The two angles above are called coterminal because they are in standard position and share the same terminal side. Angles are also coterminal when they share terminal sides as the result of complete rotations. For example, 20 degree and 380 degree angles in standard position are coterminal.

1. Measure each of the angles below. Determine three coterminal angles for each of the angles.

a. b. c.

3 Reference angles are the angle formed between the terminal side of an angle in standard position and the closest side of the x-axis. All reference angles measure between 0o and 90o.

Quadrant I Quadrant II

reference angle reference angle

Quadrant III Quadrant IV

reference angle reference angle

2. Determine the reference angle for each of the following positive angles.

a. 300o d. 210o b. 135o e. 585o c. 30o f. 870o

3. Determine the reference angle for each of the following negative angles. 4 a. -45o d. -405o b. -120o e. -330o c. -240o f. -1935o

Angles as a Part of the Unit Circle

4. The circle below is called the unit circle. Why do you believe this is so?

5. Duplicate this graph and circle on a piece of your own graph paper. Make the radius of your circle 10 squares long (10 squares = 1 unit).

6. Fold in the angle bisectors of the right angles formed at the origin. What angles result from these folds?

7. Use a protractor to mark angles at all multiples of 30o on the circle. Why didn’t we use paper folding for these angles?

8. Which of the angles from #6 have reference angles of 30o? Explain how you know.

9. Which of the angles from #6 have reference angles of 60o? Explain how you know.

10. What is the angle measure when the terminal side of the angle lies on the negative side of the x- axis? Explain how you know.

11. What is the angle measure when the terminal side of the angle lies on the negative side of the y- axis? Explain how you know.

12. What is the angle measure when the terminal side of the angle lies on the positive side of the y- axis? Explain how you know.

13. For what angle measures can the initial side and terminal side overlap? Explain how you know.

6 5.1: Homework

-250°

-290°

-140°

620°

7 380°

-195°

380°,805°

8 5.2: Right Triangles and Coordinates on the Unit Circle

1. The circle below is referred to as a “unit circle.” Why is this the circle’s name?

Part I 2. Using a protractor, measure a 30o angle with vertex at the origin, initial side on the positive x- axis and terminal side above the x-axis. Label the point where the terminal side intersects the circle as “A”. Approximate the coordinates of point A using the grid.

3. Now, drop a perpendicular segment from the point you just put on the circle to the x-axis. You should notice that you have formed a right triangle. How long is the hypotenuse of your triangle? Using trigonometric ratios, specifically sine and cosine, determine the lengths of the two legs of the triangle. How do these lengths relate the coordinates of point A? How should these lengths relate to the coordinates of point A?

4. Using a Mira or paper folding, reflect this triangle across the y-axis. Label the resulting image point as point B. What are the coordinates of point B? How do these coordinates relate to the coordinates of point A? What obtuse angle was formed with the positive x-axis (the initial side) as a result of this reflection? What is the reference angle for this angle?

5. Which of your two triangles can be reflected to create the angle in the third quadrant with a 30o reference angle? What is this angle measure? Complete this reflection. Mark the lengths of the three legs of the third quadrant triangle on your graph. What are the coordinates of the new point on circle? Label the point C.

6. Reflect the triangle in the first quadrant over the x-axis. What is this angle measure? Complete this reflection. Mark the lengths of the three legs of the triangle formed in quadrant four on your graph. What are the coordinates of the new point on circle? Label this point D.

7. Let’s look at what you know so far about coordinates on the unit circle. Complete the table.

x- y-coordinate coordinate

Notice that all of your angles so far have a reference angle of 30o.

Part II 8. Now, let’s look at the angles on the unit circle that have 45o reference angles. What are these angle measures?

9. Mark the first quadrant angle from #8 on the unit circle. Draw the corresponding right triangle as you did in Part I. What type of triangle is this? Use the Pythagorean Theorem to determine the lengths of the legs of the triangle. Confirm that these lengths match the coordinates of the point where the terminal side of the 45o angle intersects the unit circle using the grid on your graph of the unit circle.

10 10. Using the process from Part I, draw the right triangle for each of the angles you listed in #8. Determine the lengths of each leg and match each length to the corresponding x- or y- coordinate on the unit circle. List the coordinates on the circle for each of these angles in the table.

x- y-coordinate coordinate

Part III 11. At this point, you should notice a pattern between the length of the horizontal leg of each triangle and one of the coordinates on the unit circle. Which coordinate on the unit circle is given by the length of the horizontal leg of the right triangles?

12. Which coordinate on the unit circle is given by the length of the vertical leg of the right triangles?

13. Is it necessary to draw all four of the triangles with the same reference angle to determine the coordinates on the unit circle? What relationship(s) can you use to determine the coordinates instead?

14. Use your method from #13 to determine the (x, y) coordinates where each angle with a 60o reference angle intersects the unit circle. Sketch each angle on the unit circle and clearly label the coordinates. Record your answers in the table. x- y-coordinate coordinate

11 Part IV

15. There are a few angles for which we do not draw right triangles even though they are very important to the study of the unit circle. These are the angles with terminal sides on the axes. What are these angles? What are their coordinates on the unit circle? x- y-coordinate coordinate

12 5.2 : Homework Use your knowledge from lessons 5.1 and 5.2 to find the ordered pairs for each indicated angle of the unit circle.

120 60 135 45

150 30

180 0 360

210 330

225 315 240 300

270

13 5. 3: More Relationships in the Unit Circle

1. In Math II, you learned three trigonometric ratios in relation to right triangles. What are these relationships?

hypotenuse opposite

adjacent

2 Moving the triangle onto the unit circle allows us to represent these trigonometric relationships in terms of x and y. Express each of the ratios in terms of x and y.

(x, y)

3 Based on these relationships x = ______and y = ______.

4 Use this relationship to determine the coordinates of A. Both coordinates a positive. Why is this true?

b. What angle would have coordinates (-0.9397, -0.3420) on the unit circle? Why?

c. What angle would have (0.9397, -0.3420) as its coordinates? Why?

5 What is the reference angle for 250o?

a. What are the coordinates of this angle on the unit circle?

b. What 2nd quadrant angle has the same reference angle?

c. What are the coordinates of this angle on the unit circle?

6. Using a scientific or graphing calculator, you can quite easily find the sine, cosine and tangent of a given angle.

7. A student entered sin30 in her calculator and got -0.98803. What went wrong?

8. Based on the graph of the unit circle on the grid, estimate each of the values. Do not use the trig keys on the calculator for this problem. You will need to use a protractor to mark each angle and then estimate the coordinates where the terminal side of the angle intersects the unit circle.

a. cos 60o d. cos -75o b. sin 180o e. sin 490o c. tan 235o f. tan 920o

9. Use a calculator to find each of the following values. a. sin 40o e. tan 300o b. cos 40o f. cos -140o c. cos 165o g. sin 90o d. sin 165o

15 5.3: Homework

16 5.4: UnWrapping the Unit Circle – Graphs from the Unit Circle

From Illuminations: Resources for Teaching Math, National Council of Teachers of Mathematics

Materials: o Bulletin Board paper or butcher paper (approximately 8 feet long) o Uncooked spaghetti o Masking tape o Protractor o Meter stick o Colored marker o Yarn (about 7 feet long)

Part I: Unwrapping the Sine Curve

Tape the paper to the floor, and construct the diagram below. The circle’s radius should be about the length of one piece of uncooked spaghetti. If your radius is smaller, break the spaghetti to the length of the radius. This is a unit circle with the spaghetti equal to one unit.

Using a protractor, make marks every 15° around the unit circle. Place a string on the unit circle at 0°, which is the point (1, 0), and wrap it counterclockwise around the circle. Transfer the marks from the circle to the string.

Transfer the marks on the string onto the x-axis of the function graph. The end of the string that was at 0° must be placed at the origin of the function graph. Label these marks on the x-axis with the related angle measures from the unit circle (e.g., 0°, 15°, 30°, etc.).

1. What component from the unit circle do the x-values on the function graph represent?

x-values = ______

Use the length of your spaghetti to mark one unit above and below the origin on the y- axis of the function graph. Label these marks 1 and –1, respectively.

Draw a right triangle in the unit circle where the hypotenuse is the radius of the circle to the 15° mark and the legs lie along and perpendicular to the x-axis.

Break a piece of spaghetti to the length of the vertical leg of this triangle, from the 15° mark on the circle to the x-axis. Let this piece of spaghetti represent the y-value for the point on the function graph where x = 15°. Place the spaghetti piece appropriately on the function graph and make a dot at the top of it. Note: Since this point is above the x-axis in the unit circle, the corresponding point on the function graph should also be above the x-axis.

17 Transferring the Spaghetti for the Triangle Drawn to the 60° Mark

Continue constructing triangles and transferring lengths for all marks on the unit circle. After you have constructed all the triangles, transferred the lengths of the vertical legs to the function graph, and added the dots, draw a smooth curve to connect the dots.

2. The vertical leg of a triangle in the unit circle, which is the y-value on the function graph, represents what function of the related angle measure?

y-values = ______

Label the function graph you just created on your butcher paper y=sin x.

3. What is the period of the sine curve? That is, what is the wavelength? After how many degrees does the graph start to repeat? How do you know it repeats after this point?

4. What are the zeroes of this function? (Remember: The x-values are measuring angles and zeroes are the x-intercepts.)

5. What are the x-values at the maxima and minima of this function?

6. What are the y-values at the maxima and minima?

7. Imagine this function as it continues in both directions. Explain how you can predict the value of the sine of 390°.

8. Explain why sin 30° = sin 150°. Refer to both the unit circle and the graph of the sine curve.

You used the length of the vertical leg of a triangle in the unit circle to find the related y-value in the sine curve. Determine what length from the unit circle will give you the y-value for a cosine curve. Using a different color, create the graph on your butcher paper and label it y=cos x. You may want to take a picture of your graphs to reference while completing your homework. 18 5.4 Homework: Unwrapping the Cosine Curve

9. In what ways are the sine and cosine graphs similar? Be sure to include a discussion of intercepts, maxima, minima, and period.

10. In what ways are the sine and cosine graphs different? Again, be sure to include a discussion of intercepts, maxima, minima, and period.

11. Will sine graphs continue infinitely in either direction? How do you know? Identify the domain and range of y=sin x.

12. Will cosine graphs continue infinitely in either direction? How do you know? Identify the domain and range of y=cos x.

19 5.5: What is a Radian?

On a separate sheet of blank paper, use a compass to draw a circle of any size. Make sure the center of the circle is clearly marked. Use a straightedge to draw a radius of the circle.

Take a piece of string and “measure” the radius of the circle. Cut the string to exactly the length of the radius.

1. Beginning at the end of the radius, wrap the cut string around the edge of the circle. Mark where the string ends on the circle. Move the string to this new point and wrap it to the circle again. Continue this process until you have gone completely around the circle. How many radius lengths did it take to complete the distance around the circle? What geometric concept does this reflect?

2. Remember from your study of circles in Mathematics 2 that arcs can be measured in degrees or by length. In Trigonometry, we can measure arcs by degrees or radians. Based on your process in #1, what do you think a radian is?

3. Let’s consider the unit circle. We know the radius is equal to 1 unit, thus the circumference is 2 . How does this value relate to the work you did in #1?

4. So far, we know that the complete circle measures 2 r adians and 360o. Can we simplify this relationship?

5. Let’s convert several common angles from degrees to radians:

a. 180o is half of a circle, so it is how many radians?

b. 90o is a quarter of a circle, so it is how many radians?

c. 270o is three-quarters of a circle, so it is how many radians?

d. 45o is ______of a circle, so it is ______radians.

e. 120o is ______of a circle so it is ______radians.

20 6. Other angles can also be converted using the relationship between the degree measure of the angle and the associated arc length, or radian measure. Degrees to Radians Radians to Degrees

a. 32o = _____ f. = _____

b. 200o = _____ g. = _____

c. 140o = _____ h. 8 _____

d. 920o = _____ i. = _____ 12 5

e. -40 o = _____ j. 2 = _____

7. Just as you have found the values of the three trigonometric functions for specific degree measures, you will also need to find the values of these functions for radian measures. Use your knowledge of the unit circle to determine each of the following values.

a. sin = ______d. sin = ______7 4 3

b. cos = ______e. cos = ______

c. tan2 = ______f. tan = ______

8. The values of the trigonometric functions are not readily found from the unit circle. For these values, you will use a scientific or graphing calculator. Be sure your calculator is in radian model before proceeding.   a. sin = c. tan = 5 12

8 19 b. cos = d. cos = 9 10

21 5.5: Homework

22 5. 6: What’s Your Temperature?

Scientists are continually monitoring the average temperatures across the globe to determine if Earth is experiencing Climate Change. One statistic scientists use to describe the climate of an area is average temperature. The average temperature of a region is the mean of its average high and low temperatures.

1. The graph to the right shows the average high and low temperature in Atlanta from January to December. The average high temperatures are in red and the average low temperatures are in blue.

a. How would you describe the climate of Atlanta, Georgia?

b. If you wanted to visit Atlanta, and prefer average highs in the 70’s, when would you go?

c. Estimate the lowest and highest average high temperature. When did these values occur?

d. What is the range of these temperatures?

e. Estimate the lowest and highest average low temperature. When did these values occur?

f. What is the range of these temperatures?

2. In mathematics, a function that repeats itself in regular intervals, or periods, is called periodic. a. If you were to continue the temperature graphs above, what would you consider its interval, or period, to be?

b. Choose either the high or low average temperatures and sketch the graph for three intervals, or periods. .

c. What function have you graphed that looks similar to this graph?

3. How do you think New York City’s averages would compare to Atlanta’s?

23 4. Use the data in the table below to create a graph for Sydney, Australia’s average high and low temperatures.

Jan Feb March April May June July August Sept Oct Nov Dec Avg high 78 79 77 73 67 64 62 65 69 72 74 77 temp F Avg low 67 68 65 60 55 51 49 50 54 58 61 65 temp F

Use the graph to compare Sydney’s climate to Atlanta’s. What do you notice?

5. Sine and Cosine functions can be used to model average temperatures for cities. Based on what you know about these graphs from earlier units, why do you think these functions are more appropriate than a cubic function? Or an exponential function?

6. Using the data from #4, use your graphing calculator to find a sine function that models the data. Record your functions here. (Let Jan = 1, Feb = 2, etc.)

a. Average high temp:

b. Average low temp:

The a, b, c and d values your calculator reported have specific effects upon the sin graph. In the next task you will investigate the characteristics of sine, and cosine graphs. You will then revisit the functions in #6 to determine the effects of the values a – d.

24 5.6 Homework 1. The graph to the right shows the average high and low temperature in Raleigh from January to December.

a. How would you describe the climate of Raleigh, NC?

b. If you wanted to visit Raleigh, and prefer average highs in the 70’s, when would you go?

c. Estimate the lowest and highest average high temperature. When did these values occur?

d. What is the range of these temperatures?

e. Estimate the lowest and highest average low temperature. When did these values occur?

f. What is the range of these temperatures?

b. Choose either the high or low average temperatures and sketch the graph for three intervals, or periods. .

c. What function have you graphed that looks similar to this graph?

3. How do you think New York City’s averages would compare to Raleigh’s?

4. Using the data, use your graphing calculator to find a sine function that models the data. Record your functions here. (Let Jan = 1, Feb = 2, etc.)

c. Average high temp:

d. Average low temp:

5.7: Exploring Sine and Cosine Graphs

In the previous task you used your calculator to model periodic data using a sine graph. Now you will explore the sine and cosine graphs to determine the specific characteristics of these graphs.

1. Using your knowledge of the unit circle, complete the following chart for f(x)=sin x. (Use exact values.)

  x -2π -π 0 π 2π 4 2 sin x

a. What do you notice about the values in the chart?

b. When is sin x=1? sin x=0?

c. Does sin x appear to be a periodic function? If so, at what would you consider to be its period?

2. Use your graphing calculator to graph sin x. Check your mode to make sure you are using radians and make sure you have an appropriate window for your data. Make a grid below and draw an accurate graph of sin x. Make sure you draw a smooth curve.

3. Study the graph to answer the following questions:

a. What is the period?

b. What is the domain and range?

c. What is the y-intercept? d. Where do the x-intercepts occur? Page 26

e. What are the maximum values and where do they occur?

f. What is the minimum value and where does it occur?

g. How would your answers to questions d, e, and f change if your graph continued past 360?

4. Using your knowledge of the unit circle and radians, complete the following chart for f(x)=cos x

  x -2π -π 0 π 2π 4 2 cos x

5. Use your graphing calculator to graph cos x. Check your mode to make sure you are using radians and make sure you have an appropriate window for your data. Make a grid below and draw an accurate graph of cos x. Make sure you draw a smooth curve.

6. Study the graph to answer the following questions:

a. What is the period?

b. What is the domain and range?

c. What is the y-intercept?

d. Where do the x-intercepts occur?

e. What is the maximum value and where does it occur?

f. What is the minimum value and where does it occur?

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g. Picture your graph continuing past 2π. How would your answers to questions d, e, and f change?

9 h. Using your graph, find the value of cos . 4

i. Using your graph, find the value of , such that cos  0.707

7. Using your calculator, graph sin x and cos x on the same axis. How are they alike? How are they different?

8. Using what you have learned about the graphs of the sine and cosine functions, practice graphing the functions by hand. You should be able to quickly sketch the graphs of these functions, making sure to include zero’s, intercepts, maximums and minimums. Be sure to create a smooth curve!

The graphs of trigonometric functions can be transformed in ways similar to the function transformations you have studied earlier. Page 28

9. Using your knowledge of transformation of functions make a conjecture about the effect 2 will have on the following graphs of cos x and sin x. Then graph it on a calculator to determine if your conjecture is correct. Graph the parent function each time to compare the effect of 2 on the graph. (Make sure your mode is set on radians.)

a. f(x)= 2sin x f(x)= 2cos x

b. f(x)= sin x + 2 f(x)= cos x +2

c. f(x)= sin (x - 2) f(x)= cos (x -2)

d. f(x)= sin (2x) f(x)= cos (2x)

Using your conjectures from above, sketch the graph of these functions. Use your calculator to check your graphs.

e. f(x)=2 sin(x)+2

f. f(x)= 2cos(x-2)

Consider the functions f(θ)=A sin(kθ + c)+h and f(θ)=A cos(kθ + c)+h. A, k, c and h have specific effects on the graphs of the function. In trigonometry we also have special names for them based on their effect on the graph. Page 29

10. The amplitude of the function is |A|. Look back at 9a: f(x)= 2sin x and f(x)= 2cos x

a. What is the value of A? How is it related to your graphs?

b. How is the amplitude related to the distance between the maximum and minimum values?

c. What effect does A have on the effect if A<0? Graph y=-2sin x to test your conjecture.

11. The period of the function is related to the value of k. Look at 9d: f(x)=sin(2x) f(x)=cos(2x) The 2 period of the sine and cosine function is defined as , where k>0. k

a. When x was multiplied by 2, the function repeated itself twice in the usual period of 2π. What is the period of the functions in 9d?

b. Using your calculator look at the graph of the functions f(x)=sin x and f(x)=sin(4x). Notice the graph repeats 4 times in the length of time it took sin x to complete one period. So the period of f(x)=sin(4x) is 2  or , . 4 2

12. Look back at 9c. The 2 shifted the graph 2 units to the left. A horizontal translation of a trigonometric function is called a phase shift. When calculating the phase shift you have to consider the value of k, which  c determines the period of the function. The phase shift = . k If c>0, the graph shifts to the left. If c>0, the shift is to the right.

a. Graph f(x) = sin x and g(x)=sin(x+2)on the same axis using radians. For what values of x is f(x) = 0? For what values of x is g(x) = 0?

b. What do you notice about your answers to II and III?

c. Use the table feature on your calculator to investigate this relationship for values other than f(x)=0.

13. Graph y=sin x and y= cos x on the same axes. How can you use the sine function to match the graph of the cosine function? How can you use the sine function to match the graph of the cosine function?

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14. In trigonometric functions, h translates the graph h units vertically.

a. What effect does 2 have on the graph of the function y = f(x) + 2?

b. What effect did the 2 have on the graphs for 9b?

c. What happens when h > 0? h < 0 ? h = 0?

15. Look at this equation that models the average monthly temperatures for Asheville, NC. (The average monthly temperature is an average of the daily highs and daily lows.) Model for Asheville, NC where t = 1 represents January

a. Find the values of A, k, c, and h in the equation.

b. Graph the equation on your calculator. The maximum and minimum values of a periodic function oscillate about a horizontal line called the midline. What is the midline of the equation modeling Asheville’s temperature?

c. How it the value of the amplitude related to this midline?

16. How is the amplitude related to the midline of f(x)=sin x? f(x)=3sinx + 2? f(x)=-4sin x -2?

17. State the amplitude and period of the following functions describe the graph of the function.

a. f(θ)=2sin(6θ) A = __ period = ___

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b. f(θ)= -4cos(1/2 θ) A = ______period = __

The frequency of a sine or cosine function refers to the number of times it repeats compared to the parent function’s period. Frequency is usually associated with the unit Hertz or oscillations/second and measures the number of repeated cycles per second. The frequency and period are reciprocals of each other.

Example: Determine the equation of a cosine function that has a frequency of 4. 1 2 The frequency and period are reciprocals of each other so the period = . Since period = , we can replace 4 k 1 1 2 the period with and solve for k. If  , then k = 8. 4 4 k

The cosine function with a frequency of 4 is f(θ)=cos(8πθ) .

18. Find the equation of a sine graph with a frequency of 6 and amplitude of 4.

5.7: Homework

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5.8-10: Applications of Sine and Cosine Graphs

Trigonometry functions are often used to model periodic data. Let’s look at a few examples of realworld situations that can best be modeled using trigonometric functions.

1. The Flight of the Space Shuttle

As mathematicians in the US space program, you and your team have been assigned the task of determining the first orbit of the Space Shuttle on the next mission. Your project is to determine the orbit of the shuttle and any other information which might affect the remainder of the orbits during the remainder of this flight. (Will all orbits cross over the same initial points? Explain why or why not. )

Some points to remember are the shuttle may not cross land on the initial lift-off and the shuttle will launch from Kennedy Space Center in Florida.

Materials which may be used to complete this project include: Globe Poster putty String Copy of a world map Colored pencils or pens Grid paper Ruler

To begin your project: 1. Use the string to measure the circumference of a “great circle” by measuring the circumference of the globe at the equator. 2. Beginning at Kennedy Space Center, use your string to mark a great circle which represents your proposed orbit. Your orbit should alternate north and south of the equator. Hold the string in place around the globe with the poster putty. 3. Plot the coordinates of the orbit as ordered pairs (longitude, latitude) on a flat map of the globe. 4. Plot the coordinates of your orbit on graph paper. Use the intersection of the equator and the prime meridian as your origin.

Find a sinusoidal (sine) equation to fit your data. Include your coordinates for the orbit and show how you determined the equation of the orbit.

You may use a graphing calculator to check your work.

2. Musical Tones Page 33

There is a scientific difference between noise and pure musical tones. A random jumble of sound waves is heard as noise.

Regular, evenly spaced sound waves are heard as tones.

The closer together the waves are the higher the tone that is heard.

The greater the amplitude the louder the tone.

Trigonometric equations can be used to describe the initial behavior of the vibrations that give us specific tones, or notes.

a. Write a sine equation that models the initial behavior of the vibrations of the note G above middle C given that it has amplitude 0.015 and a frequency of 392 hertz.

b. Write a sine equation that models the initial behavior of the vibrations of the note D above middle C given that it has amplitude 0.25 and a frequency of 294 hertz.

c. Based on your equations, which note is higher? Which note is louder? How do you know?

d. Middle C has a frequency of 262 hertz. The C found one octave above middle C has a frequency of 254 hertz. The C found one octave below middle C has a frequency of 131 hertz.

i. Write a sine equation that models middle C if its amplitude is 0.4.

ii.Write a sine equation that models the C above middle C if its amplitude is one-half that of middle C.

iii. Write a sine equation that models the C below middle C if its amplitude is twice that of middle C.

3. The Ferris Wheel

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There are many rides at the amusement park whose movement can be described using trigonometric functions. The Ferris Wheel is a good example of periodic movement.

Sydney wants to ride a Ferris wheel that has a radius of 60 feet and is suspended 10 feet above the ground. The wheel rotates at a rate of 2 revolutions every 6 minutes. (Don’t worry about the distance the seat is hanging from the bar.) Let the center of the wheel represents the origin of the axes.

a. Write a function that describes a Sydney’s height above the ground as a function of the number of seconds since she was ¼ of the way around the circle (at the 3 o’clock position).

b. How high is Sydney after 1.25 minutes?

c. Sydney’s friend got on after Sydney had been on the Ferris wheel long enough to move a quarter of the way around the circle. How would a graph of her friend’s ride compare to the graph of Sydney’s ride? What would the equation for Sydney’s friend be?

4. Weather Models

A city averages 14 hours of daylight in June, 10 in December, and 12 in both March and September. Assume that the number of hours of daylight varies periodically from January to December. Write a cosine function, in terms of t, that models the hours of daylight. Let t = 0 correspond to the month of January.

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