Download Review – graphs, compositions and inverse trig functions

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Date__________________________
Find the amplitude, period, and frequency of the function and use this information (not
your calculator) to sketch a graph of the function over two periods. (no calc)
x
1. y = 2 cos
2. y = 20 sin 4x
3
Describe the graph of the function in terms of a basic trig function. Locate the vertical
asymptotes and graph two periods of the function. (no calc)
x
3. y = -cot 3x
4. y = 3 tan  
2
Describe the transformations required to obtain the graph of the given function from a
basic trigonometric graph. (no calc)
1
1
5. y = -3 cot   x
6. y = -2 sec   x
2
2
Solve for x in the given interval. You should be able to find these numbers without a
calculator, using reference triangles in the proper quadrants. (no calc)

3
 x
7. csc x = 2
8. sec x = - 2   x 
2
2
Graph the function for  2  x  2 adjusting the vertical window as needed. State
whether or not the function appears to be periodic. (with calc)
9. f(x) = x2 – 2cos x
10. f(x) = (1.5 cos x)2
Verify algebraically that the function is periodic and determine its period graphically.
Sketch the graph showing two periods. (no calc)
11. f(x) = cos3x
12. f(x) = |cos3x|
The graph of each function oscillates between two parallel lines. Find the equations of
the two lines and graph the lines and the function in the same viewing widow. (no calc)
13. y = 1 – 0.5x + cos 2x
Determine whether f(x) is a sinusoid. (no calc)
14. f(x) = 4cosx + 2 sin x
15. f(x) = 2 sin x – tan x
Find a, b, and h so that f(x) = a sin (b(x-h)). (with calc)
16. f(x) = cos 3x + 2 sin 3x
17. f(x) = 3 sin2x – cos 2x
Tell whether the function exhibits damped oscillation. If so, identify the damping factor
and tell whether the damping occurs as x  0 or as x   . (with calc)
18. f(x) = x sin 4x
19. f(x) =  2 cos x
Find the period and graph the function over two periods. (with calc)
20. y = cos 2x – 2 cos (3x – 1)
Graph f over the interval  4 ,4  . Determine whether the function is periodic and, if it
is, state the period. (with calc)
21. f(x) = 3x + 4 sin 2x
22. f(x) = x + sin 2x
State the domain and range of the function. (with calc)
23. f(x) = 2 – x + sin x
24. f(x) = sin |x|
Find the exact value without a calculator
 1
25. sin 1   
 2
26. cos-1 1
27. tan-1 1

3

28. cos 1  

2


29. sin(tan-1 1)
  7
30. cos 1  cos
  4
31. sin(tan-1(-1))
   
32. arccos tan   
  4 

 

Use transformations to describe how the graph of the function is related to a basic inverse
trig graph. State the domain and range. (no calc)
 x
33. g(x) = 3cos-1(2x)
34. g(x) = 3 arccos 
2
Find an exact solution to the equation without a calculator.
35. cos-1(cos x) = 1
36. tan-1 x = -1
37. From a point 100 ft from its base, the angle of elevation of the top of the Arch of
Septimus Severus, in Rome, Italy is 34  13'12". How tall is this monument? ( with calc)
38. From the top of a 100ft building a man observes a car moving toward him. If the
angle of depression of the car changes from 15  to 33  during the period of observation,
how far does the car travel? (with calc)