4 – Circular Functionsa1.phobos.apple.com/us/r30/CobaltPublic/v4/b5/3c/44/b53c444c-e10… · This...

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4 Circular Functions Problem Set 4-1 1. Convert 1 radian into degrees. 0 0 180 57.3 π 2. Convert 3.1 radians into degrees. 0 0 558 177.6 π 3. Convert π/2 radians into degrees. 0 90 4. Convert π/3 radians into degrees. 0 60 5. Convert 30 o into radians. π 6 6. Convert 45 o into radians. π 4 7. Convert 135 o into radians. 3π 4 8. Convert -270 o into revolutions. 3 revolutions clockwise 4 9. Convert 1 revolution into radians. 2π 10. The unit circle is a circle which is centered at the origin and has a radius of 1. What is true about the radian measure of an angle for a unit circle? The radian measure of an angle for a unit circle is equal to the length of the arc that it subtends. 11. What is a radian? A radian is an angle measurement that gives the ratio : length of the arc length of the radius

Transcript of 4 – Circular Functionsa1.phobos.apple.com/us/r30/CobaltPublic/v4/b5/3c/44/b53c444c-e10… · This...

Page 1: 4 – Circular Functionsa1.phobos.apple.com/us/r30/CobaltPublic/v4/b5/3c/44/b53c444c-e10… · This problem set is a two-day problem set. Problem (1) below will take an entire day.

4 – Circular Functions Problem Set 4-1

1. Convert 1 radian into degrees.

0

018057.3

π

2. Convert 3.1 radians into degrees.

0

0558177.6

π

3. Convert π/2 radians into degrees. 090

4. Convert π/3 radians into degrees. 060

5. Convert 30o into radians. π

6

6. Convert 45o into radians. π

4

7. Convert 135o into radians. 3π

4

8. Convert -270o into revolutions. 3

revolutions clockwise4

9. Convert 1 revolution into radians. 2π

10. The unit circle is a circle which is centered at the origin and has a radius of 1.

What is true about the radian measure of an angle for a unit circle? The radian measure of an angle for a unit circle is equal to the length of the arc that it subtends.

11. What is a radian?

A radian is an angle measurement that gives the ratio : length of the arc

length of the radius

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12. Consider the circle below. a) Using the radius as a scale, estimate the length of the arc in bold. The arc marked in bold is about 10 units long. (Answers may vary). b) Using your estimate from a), give the measure of marked angle in radians.

The angle in radians is length of the arc 10units

2radianslength of the radius 5units

.

r = 5

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Problem Set 4-2 1. What are the exact coordinates of the image of (1,0) after a rotation of 87o

about the origin? (cos(87o), sin(87o))

2. Estimate the coordinates of the image of (1,0) after a rotation of 87o about the origin by drawing a picture of the unit circle? Check to see if your estimate is correct with a calculator by evaluating your answer in (1). (0.05, 0.999)

3. What are the exact coordinates of the image of (1,0) after the following

rotations about the origin? a. 2π (1,0) b. π (-1,0) c. – π (-1,0) d. π/2 (0,1) e. 3 π/2 (0,-1) f. -3 π/2 (0,1)

g. π/42 2

,2 2

h. - π/42 2

,2 2

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Problem Set 4-3 1. Fill in the following table:

x -2π -7 π/4 -3 π/2 -5 π/4 - π -3 π/4 - π/2 - π/4 0

sin(x) 0 2

2 1

2

2 0

2

2 1

2

2 0

π/4 π/2 3 π/4 π 5 π/4 3 π/2 7 π/4 2π

2

2 1

2

2 0

2

2 1

2

2 0

2. Fill in the following table:

x -2π -7 π/4 -3 π/2 -5 π/4 - π -3 π/4 - π/2 - π/4 0

cos(x) 1 2

2 0

2

2 1

2

2 0

2

2 1

π/4 π/2 3 π/4 π 5 π/4 3 π/2 7 π/4 2π

2

2 0

2

2 1

2

2 0

2

2 1

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In 3 - 10, give the exact values:

3.

a. sin4

2

2

4.

a.

sin2

1 5.

a. 5

sin4

2

2

6.

a. 2

sin3

3

2

b. cos

4

2

2

b.

cos2

0

b.

5cos

4

2

2

b.

2cos

3

1

2

c.

tan4

1

c.

tan2

undef

c.

5tan

4

1 c.

2tan

3

3

7. a. osin 180

0 8.

a. osin 135

2

2

9.

a. osin 300

3-

2

10.

a.

osin 301

2

b. ocos 180

1

b. ocos 135

2

2

b. ocos 300

1

2

b. ocos 30

3

2

c. otan 180

0

c.

otan 135 1

c. otan 300

3

c.

otan 303

3

In 11 - 16, find the exact value.

11. 1

sin 1 revolutions clockwise6

3

- 2

12. 1

cos 2 revolutions clockwise12

3

2

13. 50

sin4

1

14. 50

cos4

0

15. osin 1800 0

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16. ocos 1800 1

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Problem Set 4-4 You need to be able to do 1 and 2 below: neatly, accurately, correctly, and without a calculator. Practice so that you can draw each graph in about 1 minute.

1. Sketch y = sin(x) for –2 x 2.

2. Sketch y = cos(x) for –2 x 2.

3. Sketch y = sin(x) for –2 x 4.

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4. Sketch y = cos(x) for –2 x 0.

5. Sketch y = sin(x) for 100 x 102.

6. Sketch y = cos(x) for 0 x 3

2

.

7. Extension of problem set 4-3

a. Draw a picture that includes the unit circle to explain why )0 2sin(135 =

2.

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b. Draw a picture that includes the unit circle to explain why 7

6

3cos = -

2

.

8. For the function f(x) = sin(x)

a. What is the domain? All Real Numbers

b. What is the range? –1 y 1

9. For the function f(x) = cos(x) a. What is the domain? All Real Numbers

b. What is the range? –1 y 1

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Problem Set 4-5 1. What is the definition of tan(x)?

sin x

tan x , cos x 0cos x

2. Why is the tan(/4) = 1?

π πBecause sin =cos

4 4

3. For what values of x on the interval –2 x 2 is tan(x) undefined? Explain.

3π π π 3π- , - , ,

2 2 2 2

4. Fill in the following table:

x -2π -7 π/4 -3 π/2 -5 π/4 - π -3 π/4 - π/2 - π/4 0

tan(x) 0 1 undef

1 0 1 undef

1 0

π/4 π/2 3 π/4 π 5 π/4 3 π/2 7 π/4 2π

1 undef

1 0 1 undef

1 0

5. Sketch y = tan(x) for –2 x 2.

/4

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6. Find the exact value of tan 100.5

7. For the function f(x) = tan(x)

a. What is the domain? {All Real Numbers x: x ≠ π/2 + πn}, n is any integer

b. What is the range? All Real Numbers

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Problem Set 4-6 This problem set is a two-day problem set. Problem (1) below will take an entire day. 1) Open up the “Fitting y=sin(x) & y=cos(x) to graphs” PowerPoint and follow

the instructions.

2) Practice fitting sine functions to scatterplots using the “Trig Curvefitting” Excel file. Each time you press “Press here for a new graph” on the Excel worksheet, you will get a new data set. Repeat fitting a sine function to the data until you can fit a curve to the data in less than 1 minute. Write down your estimates of h, k, a, and b before you use the Excel scrollbars.

In 3 – 6 below, complete steps a – f: Note: you can choose to use either y = sin(x) or y = cos(x) as the parent function.

a. Find an equation in y-k x-h

= sina b

(implicit) form that models the data.

b. Rewrite the equation in x-h

y = a sin +kb

(function) form.

c. Rewrite the equation in 1Y =a*sin (x-h)/b +k (calculator-ready) form. That

is, exactly what characters you would type into Y1 of your calculator.

d. Enter your equation from part c) into your calculator and graph it on the same window as the graph below. They should be the same.

e. Give the period and amplitude and explain what they mean within the context of the graph.

f. Give the coordinates of the point (h,k) and explain what it means within the context of the problem.

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3. Source: NOAA, Tides On Line, http://tidesonline.nos.noaa.gov/geographic.html, 11/7/07

a. y-5 x-7

= sin4 2

b. x -7

y = 4 sin +52

c. 1Y = 4*sin((x -7)/2)+5

e. Period – The period of the trig model is about 12.5 which means that

according to the trig model there are about 12.5 hours between one low tide and the next or between one high tide and the next. Amplitude – The amplitude of the trig model is about 4 which means that according to the trig model there is about 8 ft (2 x 4) difference between the height of ocean from high tide to low tide in Boston, MA.

f. Point – The trig model goes through the point (7,5) which means that according to the trig model, 7 hours since noon on 11/7/2007 the height of the water was 5 ft above MLLW.

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g. Using the graph below, estimate how many feet above MLLW the tide will be 24 hours since noon. About 8 feet. h. Use your equation from part b to estimate how many feet above MLLW the tide will be 24 hours since noon.

24 -7

4 sin +5 8.22

About 8.2 feet.

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4) Source: http://www.frostburg.edu/dept/phys/latta/weather/2000temp.html

Fit a model to the average daily temperature..

a. y-45 x-115

= sin25 58.1

b. x -115

y = 25 sin +4558.1

c. 1Y = 25*sin((x -115)/58.1)+45

e. Period – The period of the trig model about 365 which means that

according to the trig model there are about 365 days in the cycle of temperature swings. Amplitude – The amplitude of the trig model is about 25 which means that according to the trig model there is about 50 0F (2 x 25) difference between the average maximum temperature and the average minimum temperature.

f. Point – The trig model goes through the point (115, 45) which means that according to the trig model, on day 115 of the year, the average temperature in Frostburg, MD is about 45 0F.

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5) Source: http://www.fourmilab.ch/cgi-bin/uncgi/Earth/action?opt=-p A picture of what is light and dark on the globe at a given point. Consider the boarder between light and dark the “graph” of the function.

a. y-0 x-19,000

= sin8,000 3,981

b. x -19,000

y = 8,000 sin3,981

c. 1Y =8,000*sin((x -19,000)/3981)

a. Period – The period of the trig model is about 25,000 which means that

according to the trig model it is about 25,000 miles around the earth. Amplitude – The amplitude of the trig model is about 8,000 which means that according to the trig model it is about 16,000 (8,000 x 2) miles from the latitude in the north where there is no light in the day to the latitude in the south were there is all light in the day.

b. Point – The trig model goes through the point (19000, 0)) which means that according to the trig model 19,000 from the International Date Line, sunset was 0 miles from the equator (at the point in time when the image was created).

Miles East of the Bering Strait

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6. Source: http://www.amtsgym-sdbg.dk/as/AOL-SAT/SATURN.HTM

a. y-0 x-17

= sin200 2.7

b. x -17

y = 200 sin2.7

c. 1Y = 200*sin((x -17)/2.7)

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Problem Set 4-7 Complete the same steps a-f that you did in 4-6 for the data sets below. The following data sets are posted in Course Material 4-7: Modeling with Circular Functions 1) Average Monthly Temperature for Worcester MA

x - 4.02y = 23.3 sin +45.8

2.025

2) Deerfield Sunrise and Sunset Times

x -3.63y =184.4 sin +731

1.925

3) Refrigerator Temperatures

x -15.85y =1.03 sin +36.1

2.72

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Problem Set 4-8 Complete the same steps a-f that you did in 4-6 for the data sets below collected in class. In 1 and 2 below, complete the same steps a-f that you did in 4-6 for the data sets below collected in class. 1) Weight on a spring. 2) Pendulum

3) Open up the Fathom file, “CO2 data 1980-1990”. Fit a model to the data.

This will require you to “compose” two functions together. It is challenging, but you can do it.

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Problem Set 4-9 Source: Contemporary Precalculus Through Applications, 2

nd edition, NCSSM, Everyday Learning.

All of the problems in 4-9 come from the above source. 1. Suppose a Ferris wheel has radius 33.2 feet and makes three complete

revolutions every minute. For clearance, the bottom of the Ferris wheel is 4 feet above ground. a) Sketch a graph that show how a particular passenger’s height above

ground varies over time as he or she rides the Ferris wheel. Assume that the passenger is at the bottom of the Ferris wheel when t = 0.

b) Write a function that models the passenger’s height above ground as a function of time.

2. In Los Angeles on the first day of summer (June 21) there are 14 hours 26 minutes of daylight, and on the first day of winter (December 21) there are 9 hours and 54 minutes of daylight. On average there are 12 hours 10 minutes of daylight; this average amount occurs on March 20 and September 22. a) Sketch a graph that displays this information. Use minutes of daylight as

the response variable and number of months after June 21 as the explanatory variable.

b) Write an equation that expresses d, the number of minutes of daylight per day, as a function of t, the number of months after June 21.

3. A population of lynx oscillates in a four-year cycle. Kate is a biologist who keeps records of the lynx population in a small area. She has counted lynx on January 1 of each year. Her first count in 1994 was 40; in 1995, 60; in 1996, back to 40; and in 1997, 20. The population was back to 40 in 1998. a) Sketch a graph that shows the lynx population as a function of time.

Assume that the relationship is sinusoidal. b) Write a function that expresses the lynx population as a function of time.

4. In a tidal river, the time between high tide and low tide is approximately 6.2

hours. The average depth of the water in a port on the river is 4 meters; at high tide the depth is 5 meters. a) Sketch a graph of the depth of the water in the port over time if the

relationship between time and depth is sinusoidal and there is a high tide at 12:00 noon.

b) Write an equation for your curve. Let t represent the number of hours after 12:00 noon.