Download advanced algebra and trig - Collins Hill High School

INTEGRATED TRIGONOMETRY
FALL SEMESTER EXAM REVIEW #1
NAME_______________________________PERIOD____________DATE_____________
State the amplitude for each function.
1. y  5cos 
1.____________
2. y  2 cos 0.5
2.____________
2
3. y   sin 9
5
3.____________
State the period for each function.


4. y  3cos    
2

4.____________


5. y  6sin    
3

5.____________
  
6. y  2  sin   
 3 12 
6.____________
State the phase shift for each function.
7. y  20  5cos(3   )
7.____________
1

8. y  cos  2
4
2
8.____________


9. y  10sin   4   5
4


9.____________
10. What is the formula to determine the phase shift of a function?
10.____________
Looking at the graph, state the vertical shift of the function.
11.
12.
11.____________
12.____________
Find the values of  for which each equation is true.
13.
1
 cos 
2
14. sin   1
15. cos  0
13.____________
14.____________
15.____________
Find each value


16. sin 1  tan 
4


2
17. sin  2 cos 1

2 

18. cos(tan 1 3)
16.____________
17.____________
18.____________
19. The equation d  2.7 sin(0.5m  1.4)  12.1 models the amount of daylight in Cincinnati, Ohio, for
any given day. In this equation m=1 represents the middle of January, m=2 represents the middle
of February etc.
a. What is the least amount of daylight in Cincinnati?
a.____________
b. What is the greatest amount of daylight in Cincinnati?
b.____________
c. Find the number of hours of daylight in the middle of October.
c.____________
20. A buoy floats on the water bobbing up and down. The distance between its highest and lowest
point is 6 centimeters. It moves from its highest point down to its lowest point and back to its
highest point every 14 seconds. Write a cosine function that models the movement to the
equilibrium point.
20.____________
INTEGRATED TRIGONOMETRY
FALL SEMESTER EXAM REVIEW #2
NAME_______________________________PERIOD____________DATE_____________
Use the given information to determine the exact trigonometric value.
1
1. cos   , 0    90 ;csc 
4
2. cot  
 6 
,     ; tan 
3 2
1.____________
2.____________
Solve each equation for 0    360 .
3. 4 cos 2 x  2  0
4. sin 2 x csc x  1  0
3.____________
4.____________
5.
5.____________
3 cot x  2cos x
Use the given information to determine the exact trigonometric value.
2
6. cos    ,90    180 ;sin 
3
7. csc  
11 
,     ;cot 
3 2
6.____________
7.____________
Use a half-angle identity to find the exact value of each function.
8. cos 15
9. sin 75
8.____________
9.____________
10. tan
5
12
10.____________
Use the given information to find 10. sin2 , cos2 , and tan2 .
11. cos =
12. cos =
4
, 0    90
5
sin2 ____________
4

, 0    sin2 ____________
5
2
cos2 ____________ tan2 ____________
cos2 ____________ tan2 ____________
INTEGRATED TRIGONOMETRY
FALL SEMESTER EXAM REVIEW #3
NAME_______________________________PERIOD____________DATE_____________
Find each power. Express the result in rectangular form.
 

 
1. 3  cos  i sin 
6
6 
 
3
 

 
2.  2  cos  i sin  
4
4 
 
1.____________
5
Solve each triangle. Round to the nearest tenth.
3. b  7, c  10, A  51
2.____________
3.a=___________
B=___________
C=___________
4. a  16, c  12, B  63
4.b=___________
A=___________
C=___________
5. 4. C  79.3 , a  21.5, b  13
5.c=___________
A=___________
B=___________
Find the area of each triangle. Round to the nearest tenth.
6. a  4, b  6, c  8
6.____________
7. a  17, b  13, c  19
7.____________
8. The longest truck-mounted ladder used by the Dallas Fire Department is 108 feet long and
consists of four hydraulic sections. Gerald Travis, aerial expert for the department, indicates
that the optimum operating angle of this ladder is 60 . The fire fighters find they need to reach
the roof of an 84-foot burning building. Assume the ladder is mounted 8 feet above the ground.
a. Draw a labeled diagram of the situation.
b. How far from the building should the base of the ladder be placed to achieve the optimum
operating angle?
8b.____________
c. How far should the ladder be extended to reach the roof?
8c.____________
9. When a 757 passenger jet begins its descent to the Ronald Reagan International Airport in
Washington, D.C., it is 3900 feet from the ground. Its angle of descent is 6 .
a. What is the plane’s ground distance to the airport?
9a.____________
b. How far must the plane fly to reach the runway?
9b.____________
Simplify the expression.
10. cos x csc x tan x
10.____________
Use the unit circle to find each value.
11.____________
12.____________
11. sin 90
12. tan 360
13. cot(180)
13.____________
14. csc 270
15. cos(270)
16. sec180
14.____________
15.____________
16.____________
Find the values of sine, cosine, and tangent for each  A .
17.
B
80
A
60
C
17.
Sin=____________
Cos=____________
Tan=____________
Find the values of the six trigonometric functions for angle  in standard position if a point with
the given coordinates lies on its terminal side.
18. ( 4, 3)
18.
Sin=____________
Cos=____________
Tan=____________
Csc=____________
Sec=____________
Cot=____________
Solve each equation if 0  x  360
19. sin x  1
20. tan x   3
19.____________
20.____________
Find each value.
21. sin 1 0
22. Arc cos 0
23. Tan 1
3
3
21.____________
22.____________
Change each degree measure to radian measure in terms  .
24. 137
25. 210
26. 300
23.____________
24.____________
25.____________
26.____________
Change each radian measure to degree measure. Round to the nearest tenth.
27.
7
12
28.
11
3
29. 17
27.____________
28.____________
29.____________
INTEGRATED TRIGONOMETRY
FALL SEMESTER EXAM REVIEW #4
NAME_______________________________PERIOD____________DATE_____________
Write each rectangular equation in polar form.
1. x  7
1.____________
Graph each polar equation.
5
2.   30
3.  
4
4.   150
Express each complex number in polar form. Make sure you calculator is in degrees.
5. 6  8i
6. 4  i
7. 20  21i
5.____________
6.____________
7.____________
Write each polar equation in rectangular form.
8. r  3
9. r  2
10. r  2csc
8.____________
9.____________
10.____________
Find the polar coordinates of each point with the given rectangular coordinates.
11. (0,1)
12. (1, 3)
1
3
13. ( ,  )
4
4
11.____________
12.____________
13.____________
Find the rectangular coordinates of each point with the given polar coordinates.
1 3

14. ( , )
15. (1,  )
16. (2, 270 )
14.____________
2 4
6
15.____________
16.____________
Simplify each expression.
17. sin x cos x sec x cot x
18. cosxtanx +sinxcotx
17.____________
18.____________
19. (sin x  cos x)2  (sin x  cos x)2
20. When do you use the law of cosines? When do you use the law of sines?
19.____________
Evaluate each function for the given value.
21. f (x )   x 2  2x
find f (  2)
21.
22. f ( x )  2x 2  x
find f ( x  h )
22.
State the domain of the function, use interval notation to describe the domain.
23. f ( x) 
3 x
23.
24. f ( x) 
3x
x 2
24.
25. f ( x)  4 x  1
25.
IV.
Given the following: f ( x )  2 x 2  x  1,
g ( x )  2x
26. (f + g)(x)
26.
27. (f – g)(x)
27.
28. ( f  g ) ( x )
28.
29. (
f
)( x )
g
29.
30. ( f  g )( x )
30.
31. ( g  f )( x )
31.
List the transformations that change the parent function f ( x)  x 2 .
32.
f ( x) 
1 2
x
3
33. f ( x)  

32.
x2
2
5
34. f ( x)  (4 x) 2
V.
35.
33.
34.
Determine the intervals for which the following are increasing, decreasing, or constant.
Write your answers in interval notation.
36.
35. Increasing:
Decreasing:
Constant:
36. Increasing:
Decreasing:
Constant:
INTEGRATED TRIGONOMETRY
FALL SEMESTER EXAM REVIEW #5
NAME_______________________________PERIOD____________DATE_____________
Use the given information to determine the exact trigonometric value.
1
1. sin    ,180    270 ; tan 
3
1.____________
2
3
;cos 
2. tan   ,    
3
2
2.____________
7
3. sec    ,180    270 ;sin 
5
3.____________
4. An architect is designing a new parking garage for the city. The floors of the garage are to be
10 feet apart. The exit ramps between each pair of floors are to be 75 feet long. What is the
measurement of the angle of elevation of each ramp?
5. At a certain time of day, use angle of elevation of the sun of 44. Find the length of a shadow
cast by a building 30 meters high.
Determine the quadrant in which the angle lies. (The angle is given in radian measure)
6.
10.

_______
5
11
_______
9
7.
7
_______
5
8.
11.
7
_______
6
12.
7
________
4
13
________
4
GRAPH

2
14. f(x)  sin 
if
2  
15. f(x)  cos 
if

  4
2
16. y  4 cos

3
17. y  2sin 
9.
13.

_________
12
2
_________
3
Solve the following equations given the following stipulations.
18. Solve for 0  x  360
cos 2 x  cos x  1  sin 2 x
18.
19. Solve for 0    2
cos2   3 cos   2  0
19.
20. Solve for 0  x  360
sin x cot x  
21. Solve for all real values.
sin x  cos 2 x  1
1
2
20.
21.


Write an ordered pair that represents YZ . Then find the magnitude of YZ .
22. Y 1,6 Z  -2,5
23. Y  1,0 Z  -3,6
22.____________
23.____________
24. Write LB as the sum of unit vectors. L(0,7), B(-7,-2)
24.
25. Write ST as the sum of unit vectors. S(0,4), T( 5, 4)
25.
26. Find the unit vector having the same direction as v.
26.
v  5i  3 j
27. Write the vector v in the form, ai  bj , given its magnitude v and the angle  it makes
with the positive x-axis.
v  2 ,   135
27.