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Math Analysis CP Algebra Review For each item, find or create a sample problem and then solve it. Graphs are to be neatly completed on graph paper. 1. Given an algebraic relation, determine if it is a function. 2. Given a function, state its domain and range. 3. Given a function in function notation, evaluate it at a given value. 4. Given two coordinate points, write the equation of the line that contains them. 5. Given a linear function, determine its slope, x-intercept, and y-intercept. 6. Given a horizontal or vertical line, determine its slope, intercepts, and graph. 7. Given a direct variation, determine its slope, intercepts, and graph. 8. Given an absolute value equation, determine its slope, intercepts, and graph. 9. Given a quadratic function, determine the y-intercept, zeros, and graph. 10. Given a rational function, determine the y-intercept, zeros, and excluded values. 11. Given a linear function, find its inverse and graph both the original and inverse.
12. Given two functions, find their composition f g x .
---------------------------------------------------------------------------------------------------------------- Math Analysis CP First Day of School You have two assignments to complete. They are due on Tuesday 9/3/13. Math Analysis Diagnostic Test
Go to http://mdtp.ucsd.edu/test_new/?show_instructions=4 Take the diagnostic test. You may use a calculator. Print the results and attach it to the assignment below. You only need to complete the test for full credit. This test is for your benefit to help find areas of weakness. It would be useful to review these areas over the next few weeks.
Math Analysis Letter
You will write me a letter that addresses the items listed below. The letter must be typed and free of spelling and grammatical errors. Each bullet should be well explained using at least one paragraph.
Tell me about your mathematical career. How did you get to this point? What has been your favorite math class? Why?
How do you best learn math? What studying techniques have you utilized? What techniques have worked? Which techniques have not worked? Why?
What grade do you hope to earn in this class? What are you going to do to achieve that goal?
At the bottom of the letter, include the following information: You name and email, parent/guardian name(s), email, and phone number.
Math Analysis CP- Chapter 3A WS 3.2- Transformations of Graphs 1. For each function, graph the parent function and the given transformation on the
same coordinate grid.
a. 2
2 3 4f x x
b. 1
5f x
x
c. 7 2f x x
d. 31
1 33
f x x
e. 1
52
f x x
f. 32f x x
g. 3
2f xx
h. ( ) 3 9 3f x x
i. 2.5f x x
j. 21
6 32
f x x
2. Describe how the graphs of f x and g x are related.
a. f x x and 5 25g x x
b. 3f x x and 33
84
g x x
c. sinf x x and 5sin2g x x
d. 2xf x and 32 1.5
xf x
3. Write equations for the graphs of g x , h x , and k x if the graph of
3f x x is the parent graph.
Math Analysis CP- Chapter 3A WS 3.1- Symmetry of Graphs 1. Sketch each graph and determine whether it is symmetric with respect to the
origin.
a. 2
f xx
b. 3 1f x x c. 2 2 16x y
2. Sketch each graph and determine if it is symmetric with respect to the x-axis, y-
axis, the line y = x, or none of these.
a. 2 2y x
b. 2
1f x
x
c. 33f x x
d. 2 23.5 25x y
e. cosy x
f. 4 3f x x
3. Sketch each function to determine if it is even, odd, or neither. Then prove your
answer algebraically.
a. 22 5f x x
b. 2
5f x x
c. 3
1f x
x
d. 3 2xf x
4. Complete a sketch of the graph at right so that it is
a. An odd function b. An even function c. Neither odd nor even
Math Analysis CP- Chapter 3A WS 3.X- Families of Graphs Review 1. What is a mathematical function? Draw a graph of a function that is (a) one-to-
one, (b) many-to-one, and (c) a relation that is not a function. 2. For each function below, graph the parent function and the given transformation
on the same coordinate grid.
a.
31
2f x
x
b.
213
2f x x c. ( ) 3f x x
3. State the domain and range of each graph in problem 2.
4. Write the equations for the graphs of g x , h x , and k x if the graph of
logf x x is the parent graph. Describe each transformation.
5. Discuss the symmetry of the graph of 2 3x y .
6. Is 2
1( )f x
x an even function, odd function, or neither? Prove your answer
algebraically.
7. Use three different methods to show that 3( ) 4f x x and 3 4g x x are
inverses. 8. To convert a temperature in degrees Fahrenheit (F) to degrees Celsius (C), you
subtract 32 from F and multiply the quantity by 1.8. a. Write and graph an equation for C as a function of F. b. Find the inverse of your equation and graph it.
Analysis CP- Chapter 5A WS 5A.1- The Unit Circle
sin cos sin cos sin cos sin cos
0°
90°
180°
270°
30°
120°
210°
300°
45°
135°
225°
315°
60°
150°
240°
330°
90°
180°
270°
360°
90
180 0
270
Analysis CP- Chapter 5A WS 5.2- Trig Chart Complete the chart with the exact values of the six trigonometric functions for the given angle measurements.
Radian Degree sin θ cos θ tan θ csc θ sec θ cot θ
0°
30°
45°
60°
90°
120°
135°
150°
180°
210°
225°
240°
270°
300°
315°
330°
360° MEMORIZE THE TRIG RATIOS FOR 0°, 30°, 45°, 60°, 90°
Analysis CP- Chapter 5A WS 5.X- Chapter 5A Review Complete each problem without a calculator on a separate sheet of paper. 1. Convert each angle measurement to a decimal degree measure.
a) 2 ½ revolutions clockwise b) -57°45’ c) 13
3
2. Find the exact values of the six trig functions if 11
6
3. Verify 2cos90 2cos 45 1
4. Evaluate: a) 23cos cos sin sin2 6 3 6
b) 3csc 4sec
2 2
5. Solve for 0 360
a) cot 1 b) csc 2 c) 2sec 1 d) cos 1.5
6. Sketch each angle in standard position and give the measure of its reference angle
a) -17° b) 185° c) 5
6
7. The terminal side of angle Ө lies in quadrant II and coincides with the line y x
a) Find tan b) Find cos c) d) the reference angle
8. If the central angle of a circle is 120° and the radius is 3, find
a) the arc length b) the area of the sector 9. You are riding a Ferris wheel with radius of 60 feet. You board the ride at the bottom of the wheel which is 5 feet off the
ground. How far above the ground are you when the wheel has completed 2
13
rotations?
10. Explain why the equation 2 2sin cos 1 must be true for all values of .
------------------------------------------------------------------------------------------------------------------------------------------------------------------ Analysis CP- Chapter 5A WS 5.X- History of Trigonometry EC Answer the following questions with help from online sources. Type your answers in complete sentences.
1. What is the origin of the word “trigonometry”?
2. What branch of science first used trigonometry?
3. Who is known as “the father of trigonometry” and approximately when did he live?
4. Name one other ancient Greek who did early work with trigonometry and tell something he contributed.
5. Name one Hindu (Indian) who did early work with trigonometry and tell something he contributed.
6. Name one Islamic (Arab) who did early work with trigonometry and tell something he contributed.
7. What did Leonhard Euler contribute to trigonometry?
8. Give a brief explanation of the origins of the word “sine.”
9. How were cosecant and secant defined?
10. What does the prefix “co” mean in cosine, cosecant and cotangent?
Math Analysis CP Name __________________________ Period _______ Sine/Cosine Parent Functions
This page will help you review the graphs of the parent functions siny x and cosy x , where x is measured in
radian. You should complete this page without the aid of a graphing calculator.
Use the unit circle to help evaluate the sine function y = sin(x) for values of x that are multiples of 4
between
2 and 2 . Give exact values.
x
2
7
4
3
2
5
4
3
4
2
4
0
sin x
x
4
2
3
4
5
4
3
2
7
4
2
sin x
Now use the ordered pairs to sketch a graph of y = sin(x). Note that the angle measure, x, is measured in radian.
Use your graph to answer the following questions about the sine function.
1. Is the sine function periodic? ______________ What is its period? ________________
2. What is the domain of the sine function? ________________________________________
3. What is the range of the sine function? ________________________________________
4. Where are the x-intercepts located? __________________________________________
5. Where is the y-intercept? _____________________
6. What is the maximum value of the graph? ________ Where do the maximums occur? ____________________
7. What is the minimum value of the graph? ________ Where do the minimums occur?______________________
6.1A
Use the unit circle to help evaluate the function y = cos(x) for values of x that are multiples of 4
between 2 and
2 . Give exact values.
x
2
7
4
3
2
5
4
3
4
2
4
0
cos x
x
4
2
3
4
5
4
3
2
7
4
2
cos x
Now use the ordered pairs to sketch a graph of y = cos(x). Note that the angle measure, x, is measured in radian.
Use your graph to answer the following questions about the cosine function.
1. Is the cosine function periodic? _____________ What is its period? ________________
2. What is the domain of the cosine function? ______________________________________
3. What is the range of the cosine function? _______________________________________
4. Where are the x-intercepts located? __________________________________________
5. Where is the y-intercept? _____________________
6. What is the maximum value of the graph? ________ Where do the maximums occur? ____________________
7. What is the minimum value of the graph? ________ Where do the minimums occur?_____________________
8. We sometimes refer to the point on the graph where x = 0 as the “starting point” of the graph. What is the
starting point of the cosine graph? _____________________ Of the sine graph? ________________________
6.1B
Math Analysis CP Name __________________________ Period _______ Sine/Cosine Vertical Shift
This page will help you investigate siny x D and cosy x D . Be sure that your graphing calculator is in
Radian mode. Set the graphing window to min 2 , max 2 , min 5, max 5, scl 2, scl 1X X Y Y X Y .
Equation D Graph using 2 2x Max value
Min value
1
2Max Min
1. siny x
2. sin 2y x
3. sin 1.5y x
4. cosy x
5. cos 2.5y x
6.2A
Based on your investigation of D, answer the following questions.
1. A “default value” is the value in the parent equation, siny x . What is the default value of D? ______________
2. When 0D , what happens to the sine or cosine graph? _____________________________
3. When 0D , what happens to the sine or cosine graph? _____________________________
4. For periodic functions like the sine or cosine graph, the average value of the graph is given by 1
2Max Min .
What does the average value of siny x D tell you about the graph? _________________________________
5. What is the centerline of the parent graph cosy x ? ________________________________________________
6. The graph of cos 2y x has a new centerline because it has been shifted vertically from its original centerline,
the x-axis. What is the equation of the new centerline? _________________________________________________
7. Write a formula for the centerline of siny x D in terms of D. _______________________________________
8. In general, what effect does D have on the graph of siny x D or cosy x D ?
_______________________________________________________________________________________________
9. Graph each of the equations below without using a calculator. Then, check your answer on the calculator.
a. sin( ) 3y x b. cos( ) 2.5y x
10. Write an equation in the form siny x D from the information given below.
Maximum Value Minimum Value Vertical Shift Equation
3 1
-1 -3
4 2
2.5 0.5
6.2B
Math Analysis CP Name __________________________ Period _______
Sine/Cosine Amplitude
This page will help you investigate siny A x D and cosy A x D . Be sure that your graphing calculator is in
Radian mode. Set the graphing window to min 2 , max 2 , min 5, max 5X X Y Y .
Equation A Graph using 2 2x Max value
Min value
1
2Max Min
1. sin 1y x
2. 2sin 1y x
3. 0.5sin 1y x
4. 3cos 2y x
5. 3cos 2y x
6.3A
Based on your investigation of A and D, answer the following questions.
1. What is the default value of A? ___________________________
2. When 1A , what happens to the sine or cosine graph? _______________________________________________
3. When 1A , what happens to the sine or cosine graph? _______________________________________________
4. For periodic functions like the sine or cosine graph, the Amplitude of the function is given by 1
2Max Min .
What does the Amplitude of siny A x D tell you about the graph? __________________________________
_______________________________________________________________________________________________
5. Are the graphs of 3cos 2y x and 3cos 2y x symmetric? If so, to what line? ____________________
6. What overall effect does A have on the graphs of siny A x D and cosy A x D . Be sure to include both
the magnitude and sign of A. ______________________________________________________________________
_______________________________________________________________________________________________
7. Write formulas in terms of A and D for each of the quantities below. Remember that A and D can be positive or
negative.
Maximum value: ______________________________________________________
Minimum Value: ______________________________________________________
Amplitude: ___________________________________________________________
Equation of Centerline: _________________________________________________
11. Graph each of the equations below without using a calculator. Then, check your answer on the calculator.
a. 2sin( ) 3y x b. cos( ) 2.5y x
12. Write an equation in the form siny A x D from the information given below.
Maximum Minimum Amplitude Vertical Shift Equation
1 -3
2 -1
4 -2
-0.5 -2.5
13. Write a cosine equation in the form cosy A x D that increases over the interval 0 x and has an
amplitude between 1.0 and 2.0. _____________________________________________________________
6.3B
Math Analysis CP Name __________________________ Period _______ Sine/Cosine Phase Shift
This page will help you investigate siny A x C D and cosy A x C D . Be sure that your graphing
calculator is in Radian mode. Set the graphing window to min 2 , max 2 , min 5, max 5X X Y Y .
Equation C Graph using 2 2x Amount and Direction of Phase Shift
“x-intercepts”
1. siny x
2. sin2
y x
3. sin4
y x
4. 2cos3
y x
5.
3cos 13
y x
6.4A
Based on your investigation of A, C and D, answer the following questions.
1. What is the default value of C? _______________________________
2. What overall effect does C have on the graphs of siny A x C D and cosy A x C D ?
__________________________________________________________________________________________________
3. What effect does the sign of C have on the direction of the horizontal shift? ________________________________
__________________________________________________________________________________________________
4. What does C do to the “x-intercepts” of the graph? ____________________________________________________
__________________________________________________________________________________________________
5. Why is “x-intercepts” in quotes? (Hint: think about the graph when 0D ) ________________________________
__________________________________________________________________________________________________
6. Based on your knowledge of the period, what would cos 2y x look like? Why? _______________________
__________________________________________________________________________________________________
7. In the equation siny A x C D , in what direction do A and D move the graph? _______________________
In what direction does C move the graph? _______________________________________________________________
8. We frequently call the point (0,0) the “starting point” of siny x . Why is “starting point” in quotes?
__________________________________________________________________________________________________
What would be the “starting point” of cosy x ? _______________________________________________________
9. Which two of the three parameters A, C, and D control the starting point of the graph? _______________________
10. Graph each of the equations below without using a calculator. Then, check your answer on the calculator.
2sin4
y x
cos 22
y x
2cos 1
3y x
3sin 23
y x
6.4B
Math Analysis CP Name __________________________ Period _______ Sine/Cosine Period
This page will help you investigate siny A B x C D and cosy A B x C D . Be sure that your graphing
calculator is in Radian mode. Set the graphing window to min 2 , max 2 , min 5, max 5X X Y Y .
Equation B Graph using 2 2x Number of Cycles in 2π
Length of One Cycle
1. siny x
2. sin 2y x
3. 1
sin2
y x
4.
sin 22
y x
5.
1
cos2
y x
6.5A
Based on your investigation of A, B, C, and D, answer the following questions.
1. What is meant by Period with respect to sine and cosine graphs? _________________________________________
2. What is the default value of B in the parent functions, siny x and cosy x ? _________________________
When 1B , what is the period of the parent functions? _______________________________________________
How many cycles of the graph will you see between 0 and 2π? ___________________________________________
3. When 1B , what happens to the period of the graph? ________________________________________________
What happens to the number of cycles between 0 and 2π? ______________________________________________
4. When 0 1B , what happens to the Period of the graph? _____________________________________________
What happens to the number of cycles between 0 and 2π? ______________________________________________
5. Write a formula that shows the relationship between B and the period of the graph (measured in radian).
Remember that your formula must work for all the problems you have done. _______________________________
6. Rewrite the formula using degrees instead of radian. ___________________________________________________
7. Write an equation in the form siny Bx or cosy Bx for each period.
a. ____________________________________________
b. 2
3 ___________________________________________
c. 12 ___________________________________________
8. Write a cosine equation whose graph has amplitude 2 and period 2
______________________________________
9. Write a sine equation whose graph has a vertical shift of -2, amplitude of 1.5, and period of 4 .
_______________________________________________________________________________________________
10. Graph each of the equations below without using a calculator. Then, check your answer on the calculator.
a. 2sin 4y x b. 1
cos 2.52
y x
b. 3cos 33
y x
d. 15sin 4 56
y x
6.5B
Math Analysis CP Name __________________________ Period _______ Sine/Cosine Graphs
1. Based on your investigation, what overall effect does each have on the graphs of siny A B x C D and
cosy A B x C D
A: _______________________________________________________________________________________________
B: _______________________________________________________________________________________________
C: ________________________________________________________________________________________________
D: _______________________________________________________________________________________________
2. Graph each of the following equations without a calculator. Then check your answers.
Equation Center-
line
Amp-
litude
Period Phase
Shift
Graph
2sin3
y x
cos 3 2y x
3sin 12
y x
cos( ) 3y x
6.6A
Equation Center-
line
Amp-
litude
Period Phase
Shift
Graph using 2 2x
3cos 2y x
3sin 2 1y x
22cos 3
3y x
2cos 90 1y x
Note that x is in degrees
3. Write an equation of the form siny A B x C D for the information below.
Maximum Minimum Period Phase Shift Equation
3
-2 2 0
1 -1
2
4
2 0 0
1 -3
2
0
6.6B
Math Analysis CP Name __________________________ Period _______ Sine/Cosine Graphs
For each graph, determine the values of D, A, B and C. Then, write an equation in the form siny A B x C D
and cosy A B x C D .
Sine Equation Cosine Equation
1. A = B = CSin = CCos = D =
2. A = B = CSin = CCos = D =
3. A = B = CSin = CCos = D =
4. A = B = C = D =
6.7A
Sine Equation
Cosine Equation
5. A = B = CSin = CCos = D =
6. A = B = CSin = CCos = D =
7. A = B = C = D =
8. A = B = CSin = CCos = D =
6.7B
Math Analysis CP Name __________________________ Period _______ Tangent and Cotangent Graphs
This page will help you investigate tany A B x C D and coty A B x C D .
Recall that
sintan
cos
xx
x
1. Graph cosy x on your calculator.
For what values of x does cos 0x ?
______________________________________________
These values must be excluded from tany x . Why?
______________________________________________ The excluded values will be vertical asymptotes of the tangent graph. Draw them as dashed lines on the graph.
2. Graph siny x on your calculator. Where does the sine graph cross the x-axis? ______________________
Why will the tangent graph cross the x-axis wherever the sine graph crosses it? _________________________ _______________________________________________________________________________________________ Draw the appropriate points on the graph above.
3. Use you calculator to graph tany x . Sketch your graph above. Notice the relationship between your answers
to (1) and (2) and to the tangent graph.
4. Is the tangent graph periodic? _____________________ What is its period? ______________________________ Does the tangent graph have a maximum or minimum? _______________ If so, what are they? _______________ Does the tangent graph have a “centerline”? __________________ If so, what is it? _________________________ Why is “centerline” in quotes here? _________________________________________________________________
5. Use your calculator to graph
sin 2y x and tan 2y x
Where does the sine graph cross the line 2y ? __________
Where does the tangent graph cross 2y ? ______________
Compared to the graph of tany x , each point of
tan 2y x is shifted _____________________________
Overall, what effect does D have on the graph of tany x D ________________________________________
6. Use your calculator to graph 2 tany x .
How does the graph compare to tany x ?
(Hint: you might want to look at the TABLE values) _________________________________________________
What effect does A have on the graph of tany A x ?
_________________________________________________
6.8A
7. Graph tan2
y x
What effect does C have on the graph of tany x C ?
_____________________________________________
8. Graph tan 2y x
What is the period of the graph? __________________ Write a formula for the period of a tangent graph. ______________________________________________
Use your formula to predict the period of 1
tan2
y x
_____________________________________________ Check your prediction using the calculator. Were you correct? ___________
9. In general, do your concepts of A, B, C, and D for the sine graph hold true for the tangent graph? _______________
10. Recall that
cos1cot( )
tan sin
xx
x x
Where will the asymptotes of the cotangent graph be? ____________________________________________ Where will the graph cross x-axis? ____________________________________________ Use your calculator to draw a sketch of the cotangent graph.
11. Sketch each of the equations below without a calculator. Then check your answers.
Tangent Graph Cotangent Graph
2tan 3y x
2cot 3y x
3tany x
3coty x
1tan
2y x
1cot
2y x
6.8B
Math Analysis CP Name __________________________ Period _______ Secant and Cosecant Graphs
This page will help you investigate secy A B x C D and cscy A B x C D .
Recall that
1sec
cosx
x
1. Graph cosy x on your calculator.
What values of x must be excluded from secy x .
______________________________________________ The excluded values will be vertical asymptotes of the secant graph. Draw them as dashed lines on the graph.
2. Graph secy x on your calculator and on the graph above.
3. Use you calculator to graph secy x and cosy x . What is the relationship between the maximum and
minimum values of the cosine graph and the graph of secy x ? _______________________________________
_______________________________________________________________________________________________
4. Is the secant graph periodic? _____________________ What is its period? _______________________________ Does the secant graph have relative maximums and minimums? __________________________________________ Why do we call the maximums and minimums “relative”? _______________________________________________
5. Use your calculator to graph
cos 2y x and sec 2y x
Could you have predicted the location of the asymptotes and relative maximums and minimums of the secant graph from the cosine graph? _________________________________ (If not, check that you have entered your equations correctly)
6. Use your calculator to graph 2cosy x and 2secy x
Could you have predicted the location of the asymptotes and relative maximums and minimums of the secant graph from the cosine graph? _____________________________________
7. Observe the graphs of cos( )y x and secy x . Do your graphs agree with your predictions? _______
8. Observe the graphs of cos 3y x and sec 3y x . Do your graphs agree with your predictions? ____________
9. In general, the easiest way to graph secy A B x C D is to first graph the related _____________________
function.
6.9A
10. Recall that 1
csc( )sin
xx
Where will the asymptotes of the cosecant graph be? ____________________________________________ Where will the relative minimums and maximums be? ____________________________________________
Use your calculator to graph siny x and cscy x
11. Use your calculator to sketch the graph of
2sin 2 22
y x
.
Use the sine graph to sketch 2csc 2 22
y x
Use your calculator to check that your cosecant graph is correct.
12. In general, do your concepts of A, B, C and D from the sine and cosine graphs hold true for the secant and cosecant graphs? _______________________________________________________________________________________
13. Sketch each of the equations below without a calculator. Then check your answers.
Secant Graph Cosecant Graph
2sec 1 y x
2csc 1 y x
3secy x
3cscy x
sec 2 1y x
csc 2 1y x
6.9B
Math Analysis CP Name __________________________ Period _______ Review of Trig Graphs Complete the table below. Then graph each of the equations.
1.
4sin 3 1y x 2.
5tan 2 3y x 3. 2sec 1y x
Period
Horizontal Shift
Vertical Shift
Amplitude
Maximum
Minimum
Domain
Range
Number of cycles from 0 to 2π
Intervals from 0 to 2π where the graph is increasing
1.
2.
3.
6.10A
Math Analysis CP Name __________________________ Period _______ Review of Trig Graphs Complete the table below. Then graph each of the equations.
4.
3cos 2y x 5. cot
3y x
6.
csc 2y x
Period
Horizontal Shift
Vertical Shift
Amplitude
Maximum
Minimum
Domain
Range
Number of cycles from 0 to 2π
Intervals from 0 to 2π where the graph is increasing
4.
5.
6.
6.10B
Analysis CP- Chapter 5B WS 5B.5- Right Triangle Trigonometry Solve all problems neatly on a separate sheet of paper. Diagrams must be drawn and labeled. 1. A man drives 500 feet along a road which is inclined 20 degrees to the horizontal. How high is he above his starting
point? 2. The base of a tree bent over by the wind forms a right angle with the ground. If the bent part of the tree makes an
angle of 50° with the ground and if the top of the tree is now 20 feet high, how tall was the tree? 3. Two straight roads intersect to form an angle of 75°. Find the shortest distance from one road to a gas station on the
other road 100 feet from the junction. 4. Two buildings with flat roofs are 60 feet apart. From the roof of the shorter building, 40 feet in height, the angle of
elevation to the edge of the roof of the taller building is 40°. How high is the taller building? 5. A ladder with its foot in the street makes an angle of 30° with the street when its top rests on a building on one side of
the street and makes an angle of 40° with the street when its top rests on a building on the other side of the street. If the ladder is 50 feet long, how wide is the street?
6. Find the perimeter of an isosceles triangle with a base of 40 inches and base angle of 70°. Find the missing measures of the right triangles: 7. 8. Give exact values for the lengths of all line segments (a – e) and the measures of all angles (A-G) . Then find exact values for the sine, cosine and tangent of each angle. No calculators! Note: The figure is not drawn to scale…it never is!
C A
B
35°10’
72.5 ft
A C
B
15.25 ft
32.68 ft
G
F
E D
C
B
A
e
d
c
b
a
1
2
2
1
H
Analysis CP- Chapter 5B WS 5B.7- Which Law For each triangle, (1) State the geometric theorem, (2) State which law (if any) could be used to solve the triangle, (3) Give the number of solutions and explain why. Do all work neatly on a separate sheet of paper.
1. 50 , 10, 100A c B
2. 25 , 8, 160A c B
3. 75 , 19, 45A a C
4. 105 , 75 , 25c B c
5. 6, 7, 1a b c
6. 5, 9, 11a b c
7. 100 , 50 , 30A B C
8. 65 , 12 , 9B C a
9. 15.7, 51 30', 18.3a B c
10. 26, 24, 58a b A
11. 6, 8, 150a b A
12. 25 , 125, 150A a b
Select 5 (solvable) triangles from the above and give a complete solution. 13. One angle of a rhombus is 38°42’ and its sides are 4.836 in. long.
a) Find the length of the shorter diagonal. b) Find the area of the rhombus.
14. The radius of a circle is 12 in. What is the measure of the central angle subtended by a chord 18 in. long? ----------------------------------------------------------------------------------------------------------------------------- ------------------------------------- Analysis CP- Chapter 5B WS 5B.X- Chapter 5B Review
For problems #1-6, assume angles A, B, and C are opposite sides a, b, and c in ABC
1. Given a = 5, b = 8, and C = 70°, find c. 2. Given c = 4, a = 6, and A = 50°, find C. 3. Given A = 31°20’, C = 65°50’ and c = 6, find b.
4. Given b = 6, a = 3, and A = 64°, find the area of ABC .
5. Given c = 6, B = 61°40’, and A = 92°30’, find the area of ABC .
6. Determine the number of triangles that exists and state the reason why. Then solve the triangle. If no triangle exists, give the reason why not.
a) B = 40°, b = 30, c = 20 b) B = 140°, c = 30, b = 20 c) C = 55°10’, b =480, c = 428 d) A = 60°, a = 1.5, b = 2 e) A = 30°, B= 60°, C = 90° f) a = 10, b = 4, c = 5
7. The measures of the sides of ABC are in the ratio of 15:13:7. Find the measure of the largest angle.
8. A triangular plot of ground has two sides 185’ and 147’ which intersect at an angle of 51°10’. Find the length of the third side and the area of the plot.
9. Three circles with radii of 115 ft, 150 ft, and 225 ft. are tangent to each other. Draw the triangle formed by their centers. Find the measure of the three angles and the area of the triangle.
10. A plane flying due east at 100 m/s is also being blown due south at 40 m/s by a strong wind. Find the speed and bearing of the plane.
11. In Washington D.C. the Lincoln Reflecting Pool is due west of the Washington Monument. A tourist at the top of the Washington Monument sites the front edge of the Pool at an angle of depression of 27° and sites the far edge of the pool at an angle of depression of 10°. If the Pool is 2030 feet long, about how high is the Washington Monument?
------------------------------------------------------------------------------------------------------------------------------------------------------------------ Directions for the Chapter 5 Online Test 1. Go to www.amc.glencoe.com. Choose "Chapter Test" under "Chapter Resources". Then choose the Chapter 5 Test. 2. Take the 16 question test and earn at least a 90%. You may retake the test until you earn a 90%. 3. Once you earn at least a 90%, print the entire FIRST page of results showing your score at the top000. The entire
print job should be about four pages, but I only need the first one.
Analysis CP- Chapter 7A WS Trig Identities 01- Algebra Review
Section A: Factor the following completely.
1. 2 6x x 2. 2sin sin 6x x 3. 2 22x xy y 4. 2 2sin 2sin cos cosx x x x
5. 4 4x y 6. 4 4sin cosx x 7. 3 3x y 8. 3 3sin cosx x
9. 2x xy 10. 2sin sin cosx x x 11. 2xy x 12. 2cos sin cosx x x
Section B: Simplify the following completely.
13. 2 2x y
x y
14.
2 2tan sec
tan sec
x x
x x
15.
1 1
x y 16.
1 1
sin cosx x
17. x y
y x 18.
cos sin
sin cos
x x
x x 19.
1x
x 20.
1sin
sinx
x
21. 1
xx
22. 1
tantan
xx
23. 2
3
x
x y
x xy
x
24.
2
tan
sin cos
sin sin cos
cos
x
x x
x x x
x
Section C: Complete the square.
Section D: Simplify the left side until it equals the right side.
25. 2
1 2 ___ _______x 26. 1 2 2y xxy x y
x y
27. 2
1 2sin ___ _______x 28. 1sin cos
sin coscos sin
x xx x
x x
----------------------------------------------------------------------------------------------------------------------------- ------------------------------------- Analysis CP- Chapter 7A WS Trig Identities 02- Simplifying
Completely simplify each problem. The answer to each problem will be one of the following: sin x , cos x , tan x , or 1.
1. 2 2
1 1
sec cscx x 2. 2 2sec 1 sinx x
3. sec sin tanx x x 4. sin cos tan cot sec cscx x x x x x
5. sin secx x 6. 2 2csc 1 cosx x
7. 1 tan
1 cot
x
x
8.
2
2
csc 1
cot
x
x
9. 2 21 tan 1 sinx x 10. sec
tan cot
x
x x
11.
sin tan
tan csc cot
x x
x x x
12. 4 2 2 4cos 2cos sin sinx x x x
----------------------------------------------------------------------------------------------------------------------------- ------------------------------------- Analysis CP- Chapter 7A WS Trig Identities 03- Proofs
Prove the following. Use a two-column format, showing all work.
1. 2 2
1 11
sec cscx x 2. 2 2 2cos tan cos 1x x x
3. 21 cos
csc cot1 cos
xx x
x
4.
sec sincot
sin cos
x xx
x x
5. sin 1
sec tan cot
x
x x x
6. csc sec cot tanx x x x
7. 2 2sin cot 1 sin 1 sinx x x x 8. cos cos
2sec1 sin 1 sin
x xx
x x
9. cos 1 sin
1 sin cos
x x
x x
10.
sec 1 tan
tan sec 1
x x
x x
Analysis CP- Chapter 7A WS Trig Identities 04- Sum, Difference, and Double Angle Identities
Section A: Simplify the following.
1. cos 270 x 2.
1 cos 2
sin 2
3. tan 45 x 4. sec 270
Section B: Prove each of the following.
5. 2csc 2
cscsec
6. 21
tan sin2 1 cos2
7. sin 30 cos 60 cosy y y 8. cot csc 2 cot 2
9. 2
sin cos sin 2 1 10. tan 1
tan 135tan 1
xx
x
----------------------------------------------------------------------------------------------------------------------------- ------------------------------------- Analysis CP- Chapter 7A WS Trig Identities 05- Extra Practice
Section A: Choose any 10 problems and prove the identity. Use a two-column format, showing all work.
1. tan sin sec 2. cot cos csc 3. 2 2cot csc 1
4. tan cot sec csc 5. cot
csccos
6.
2 22
2
sin cossec
cos
7. 4 4 2 2csc cot csc cot 8. 4 4 2 2sec tan tan sec 9. 2 21 tan sec 2tan
10. 2 21 sin 1 tan 1 11. 2cos
sin cscsin
12.
22
2
1 costan
cos
13. 2
tansec csc
1 cos
xx x
x
14. 2cot sec
sec csccos cot
15. 2 22sin 1 2cos 1
Section B: Choose any 6 problems and prove the identity. Use a two-column format, showing all work.
16. cos sin
1 tancos
17.
21 sin sincos tan
cos
x xx x
x
18. 2tan tan cot sec
19. sec tan sec tan 1 20.
3
cos 1 1 coscsc
sin
21.
sincot csc
1 cos
22. tan cot
sin cossec csc
23. 2sin sin
2tancsc 1 csc 1
x xx
x x
24. 2
21 2sin 3sin 1 sin
cos1 3sin
x x xx
x
25.
3 3
2
sin cos sec sin0
tan 11 2cos
------------------------------------------------------------------------------------------------------------------------------------------------------------------ Analysis CP- Chapter 7A WS Trig Identities 06- Review
Prove each problem is an identity. Use a two-column format, showing all work.
1. sec sin tanx x x 2. sin cos cot cscx x x x 3. cos csc cotx x x
4. tan sin cot cos secx x x x x 5. 2 2sec 1 sin 1x x 6. 2 2 22cos sin 1 3cosx x x
7. 2 2csc 1 cos 1x x 8. sin tan cos secx x x x 9. sin sec csc tan 1x x x x
10. 2 2cos 1 tan 1x x 11. cos csc sec cot 1x x x x 12. sec cos sin tanx x x x
13. sin cot cos
2cotsin
x x xx
x
14.
1 2sin cossin cos
sin cos
x xx x
x x
15.
22
2
1 tancsc
tan
xx
x
16. 4 4 2 2cos sin cos sinx x x x 17. 2
2
2
1 sincot
1 cos
xx
x
18.
1 sin cos
cos 1 sin
x x
x x
19. cos 60 sin 30A A 20. 1 tan
tan 451 tan
xx
x
21.
1csc 2 sec csc
2
22. cos2 1
cos 12 cos 1
xx
x
23.
cos2cos sin
cos sin
AA A
A A
24.
1 sec sin1 sin2
2 sec
A AA
A
Analysis CP- Chapter 7B WS Solving Trigonometric Equations A Give the principal solution(s) for each equation. You must write an inverse equation and show algebraic support for every answer. Answer #1-8 in Radians and #9-16 in Degrees. Give exact solutions.
1. 2sin 3 0x 2. 3 tan 1 0x 3. 2 cos 1 0x 4. 3 tan 1 0x
5. 2sin 2 0x 6. 2cos 1 0x 7. 24sin 1x 8. sin cos sin 0x x x
9. 2 23sin cos 0x x 10. 22cos 3cos 1 0x x 11. sin 2csc 1x x 12. 22 sin 2sin 0x x
13. cos2 sin 1x x 14. sin2 cos 0x x 15. 4tan sin2 0x x 16. sin2 2sinx x
Analysis CP- Chapter 7B WS Solving Trigonometric Equations B For problems #1-8, provide all real solutions (in Radian) for each equation. For #9-16, provide all solutions in the interval
0 360x . You must write an inverse equation and show algebraic support for every answer. Give exact solutions.
1. 2sin 1 0x 2. 24cos 1 0x 3. tan 3 0x
4. 24sin 4sin 1 0x x 5. 22cos 3cos 2 0x x 6. cos2 cosx x
7. 2 5cos cos2
4x x 8. 2 3
cos cos24
x x 9. cos 1x
10. sin 0.5x 11. 2sin cos cos 0x x x 12. cos2 cos 0x x
13. sin2 sin cos2 cos 1x x x x 14. 2 2sin cos cos 1 0x x x
15. cos cos 14 4
x x
16.
2sin sin
4 4 2x x
Analysis CP- Chapter 7B WS Solving Trigonometric Equations C Solve each equation for principal values of x. Express solutions in degrees. Give exact values.
Solve each equation for 0 360x . Give exact values.
1. cos 3cos 2x x 2. 22sin 1 0x 3. 2 sin 2sin cos 0x x x 4. cos2 3cos 1 0x x
Solve each equation for 0 2x . Give exact values. Solve each equation for 0 2x . Round your answers to
the nearest thousandth.
5. 24sin 4sin 1 0x x 6. cos2 sin 1x x 7. 26cos 9cos 1 0x x 8. cos 02
xx
Solve each equation for all real values of x (in Radian). Give exact values.
9. 3cos2 5cos 1x x 10. 22sin 5sin 2 0x x 11. 23sec 4 0x 12. tan tan 1 0x x
Analysis CP- Chapter 7B WS Solving Trigonometric Equations D Write the equation for the inverse of each function. Then graph the function and its inverse on their own coordinate planes. Then state the domain and range of each. 1. Sin( )y x 2. Cos( )y x 3. Tan( )y x
Evaluate:
4. Arctan( 1) 5. 1Sin 1 6. 1Cos tan4
Use your calculator to solve by graphing over the interval 0 2x
7. 2sin 1 0x 8. sin 2cosx x
Analysis CP- Chapter 09 WS 09- DeMoivre's Theorem Find the polar coordinates of the point with the given rectangular coordinates.
1. 2, 2 2. 3,1
Find the rectangular coordinates of the point with the given polar coordinates.
3. 3, 225 4. 4,1.4
Simplify.
5. 93i 6. 3 5 3 2i i 7. 6 2
2
i
i
Express each complex number in polar form.
8. 4 4i 9. 5 10. 6 6 3i
Express each complex number in rectangular form.
11. 2 cos60 sin60i 12. 2 cos 45 sin 45i
Find each product or quotient. Then write the result in rectangular form.
13. 4 cos270 sin270 3 cos45 sin45i i
14.
2 3 cos120 sin120
3 cos60 sin60
i
i
Solve each of the following using DeMoivre's Theorem.
15. 8
1 i 16. 6 1 0x
Analysis CP- Chapter 09 Multiple Choice Review
1. Simplify: 14 232 3 5i i
A) 1 3i B) 3 i C) 1 2i D) 1 i
2. Simplify: 2
5 3i
A) 16 30i B) 34 30i C) 16 30i D) 34 30i
3. Simplify: 3 2
4 5
i
i
A) 2 23
41 41i B)
22 23
41 41i C)
2 23
9 9i D)
2 23
41 41i
4. Express 5 3 5i in polar form.
A) 11 11
10 cos sin6 6
i
B)
11 1110 cos sin
6 6i
C) 11 11
5 cos sin6 6
i
D)
5 510 cos sin
3 3i
5. Express 3 3
4 cos sin4 4
i
in rectangular form.
A) 2 2i B) 2 2 2 2i C) 2 2 2 2i D) 2 2 2 2i
6. Simplify 3
2 2i and express the result in rectangular form.
A) 16 16i B) 16i C) 16 16i D) 16 16i
7. Simplify 3
3 3 3i
and express the result in rectangular form.
A) 216i B) 1
216i C)
1
216i D) 216i
8. Which of the following in not a root of 3 1 3z i to the nearest
hundredth?
A) 0.22 1.24i B) 0.97 0.81i C) 1.02 0.65i D) 1.18 0.43i
For exercises 9-10, let 1
2 28 cos sin
3 3z i
and 2 0.5 cos sin
3 3z i
.
9. Write the rectangular form of 1 2z z .
A) 4i B) 4 C) 4 4i D) 4
10. Write the rectangular form of 1
2
z
z.
A) 8 8 3i B) 8 8 3i C) 16 16 3i D) 8 8 3i
Math Analysis CP- Chapter 08 WS 8.5- Vector Applications
For each problem, make a sketch to show the given vectors and the resultant vector. Then solve the problem. Round magnitudes to the nearest hundredth and directions to the nearest tenth. DO NOT ROUND UNTIL THE FINAL ANSWER! 1. Find the magnitude and direction of the resultant of a 425-newton force along the x-axis and a 390-newton force
perpendicular to it.
2. Find the magnitude and direction of the resultant of a 105-newton force along the x-axis and a 110-newton force at an angle of 50° to the first force.
3. Two soccer players kick the ball at the same time. One player’s foot exerts a force of 70N west and the other’s foot exerts a force of 30 N south. Find the magnitude and direction of the resultant force on the ball.
4. A hiker leaves camp and walks 13 km due north. The hiker then walks 15 km northeast (exactly midway between north and east). Find the hiker’s direction and displacement from her starting point.
5. An airplane flies due west at 240 km/h. At the same time, the wind blows it due south at 70 km/h. Find the plane’s resultant velocity and direction.
6. A pilot flies a plane east for 200 km, then 60° south of east for 80 km. Find the plane’s distance and direction from the starting point.
7. Find the magnitude and direction of the resultant of two forces of 250 pounds and 45 pounds at angles of 25° and 160° with the x-axis, respectively.
8. An airplane flies at 150 km/h and heads 30° south of east. A 40 km/h wind blows it in the direction 30° west of south. What is the plane’s resultant velocity and bearing?
9. What is the force required to push a 100-pound object along a ramp that is inclined 10° to the horizontal?
10. Three vertices of a parallelogram are located at A(1, 2, 3), B(2, -1, 1) and C(-2, 1, -1). Find the area of the parallelogram.
Answers: (1) r = 576.82N, Ө = 42.5° (2) r = 194.87N, Ө = 25.6° (3) r = 76.16N, Ө = 23.2° south of west (4) r= 25.88 km, Ө = 24.2° east of north (5) r = 250 km/h, Ө = 16.3° north of west (6) r = 249.80 km, Ө = 16.1° south of east (7)r = 220.49 lb, Ө = 33.3° (8) r = 155.24 km/h, Ө = 134.9° (east of north) (9) 17.36 lb (10) Area = 17.32 square units. ----------------------------------------------------------------------------------------------------------------------------- ------------------------------------- Math Analysis CP- Chapter 08 WS 8.X- Vector Review
1. Draw vector a with magnitude 2.4 cm and direction 35° and find the magnitude of the horizontal and vertical
components of a .
2. Given A(-3, 7) and B(-4, 9) find:
a) an ordered pair for AB b) the magnitude of AB c) the direction of AB
3. Given 3, 1v and 0,4w :
a) Find 2u v w b) Find u c) True or False? 2u v w
4. Write each of the following vectors in the form ai + bj a) The vector joining the origin to P(2, -3) b) The vector joining P1(2, 3) to P2(4, 2) c) The vector joining P2(4, 2) to P1(2, 3) d) The unit vector in the direction 3i + 4j e) The vector having magnitude 6 and direction 120°
5. Given 4, 1,0v and 2,1,3w , find:
a) w b) The sum of v and w
c) The cross product of v and w d) The dot product of v and w
6. Determine if the vectors 4,1, 6 and 8, 2,5 are perpendicular.
7. Find a vector perpendicular to the plane containing the points (1, 2, 3), (-4, 2, -1) and (5, -3, 0). 8. One force of 100 N acts on an object while another force of 80 N acts on the object at a 55° angle from the first force.
Find the magnitude and direction of the resultant force. 9. An airplane flies due west at 260 km/hr. At the same time, the wind blows it 10° east of south at 75 km/hr. Find the
plane’s resultant velocity and direction. 10. Three vertices of a parallelogram are located at A(1,2,3), B(2, -1,5) and C(4,1,3). Find the area of the parallelogram.
