Download Comments FE II Review MAT187

Study Guide for PART II of the Spring 2015 MAT187 Final Exam.
NO CALCULATORS are permitted on this part of the Final Exam. This part of the Final exam will consist of 50 multiple
choice questions. You will be provided with two sheets of scratch paper which must both be turned in with you exam
regardless of whether or not you use the scratch paper. You may NOT use your own scratch paper.
This portion of the exam covers the Trigonometry portion of this course (4.1 – 4.12). You should focus on your notes
and homework from those sections and Tests #5 and #6. Of course some of you may benefit from utilizing the
supplemental reading and videos on the class Help page from these sections too!
BE SURE THAT YOU HAVE LOOKED AT, THOUGHT ABOUT AND TRIED THE SUGGESTED PROBLEMS ON THIS REVIEW
GUIDE PRIOR TO LOOKING AT THESE COMMENTS!!!
Here is a brief overview of what you should be able to do!
1. Be able to convert a degree measure into a radian measure (where  will be part of your answer). Try 2550
2250 

180
0

5
radians
4

2. Be able to evaluate a trig function at a particular degree measure. Example: cos 1200

1
 just use the unit circle!
2
3. Be able to utilize you unit circle to find the value of trig expressions like cos
cos
4

 sin
3
2
4

1
3
 sin    1   again, just use the unit circle!
3
2
2
2
4. Know which trig functions are positive and which ones are negative in each quadrant!
S
A
T
C
5. Given a standard angle on the unit circle, be able to identify the coordinates.
Having a solid understanding of the unit circle will help you on MANY of these problems!



6. Be able to find the value of an inverse trig function expression. Example: arctan  
3
 . Be sure to practice arcos
3 
and arcsin problems from your notes and homework too.

3



and go find the only place where the tangent of that
arctan  
   draw the partial unit circle from  to
2
2
6
 3 
angle is 
3

11
, note the answer is  NOT
6
6
3






7. Be able to find the value of composed trig functions and inverse trig functions. Example: sin  arccos  
Find others in your notes and homework to practice!


3 
 5  1
sin  arccos  
   sin   

 6  2
 2 

3 
 .
2  
8. Be able to evaluate various trig functions at certain degree measures. Example:
cos300 ,  cos 2100 ,sin 600 ,  sin 600
Answers are
3
3 3
3
respectively
,
,
,
2
2 2
2
9. Be able to calculate the length of an arc when given the radius and angle (in radians). See section 4.4
S  r ,  in radians!
10. Can you solve tan x  cot x ? Think about the unit circle!
Since tan  
x
y
and
cot  
y
we just need to find any places on the unit circle where those quotients of the
x
coordinates match!
11. Be able to simplify a basic trigonometric expression. Example:
cot 
csc 
cos 
cot  sin  cos  sin 



 cos 
1
csc 
sin 
1
sin 
12. Can you simplify 1  sin x 1  sin x  ?
1  sin x 1  sin x   1  sin x  sin x  sin 2 x  1  sin 2 x  cos2 x
13. What is sin 600 cos 400  cos 600 sin 400 equivalent to? cos1000 ,sin1000 ,cos 200 ,sin 200 .
sin 600 cos 400  cos 600 sin 400  sin  600  400   sin 200 Make sure that you know the trig identities that I told you
were important! You should recognize THIS example as an example of the sine of the difference of two angles. Be sure
to think about what this problem would look like if it led to the sine of the sum of two angles or the cosine of the sum or
difference of two angles.
3
5
and cos B 
and A and B are both acute angles what does cos  A  B   ?
14. If
5
13
What about sin  A  B  ?
sin A 
To solve this problem draw two right triangles, one for angle A and one for angle B. On the triangle with angle A in it you
can solve for the missing side and get x = 4 and the one with angle B in it and get y = 12. Then just remember the correct
sum and difference formulas.
4 5 3 12 20 36
16
cos  A  B   cos A cos B  sin A sin B     


5 13 5 13 65 65
65
3 5 4 12 15 48
33
sin  A  B   sin A cos B  cos A sin B     


5 13 5 13 65 65
65
15. Be able to simplify trigonometric expressions involving double angles. (go look some up!)
Be sure that you are familiar with the double angle identities for sine and cosine (#20 and #21). Be sure that you know
all three versions of #21!
16. Be able to solve basic trig equations in restricted domains. Example: Solve tan x  3  0 in
1800 ,3600 
tan x  3  0  tan x   3  x  3000 Use the unit circle on the given interval to look up the only place
where y/x is  3
 
 
17. Given a modified Sine equation be able to figure out the amplitude. y  A sin  B  x 
C 
 D
B  
Amplitude = A
18. Given a modified Sine equation be able to figure out the period.
Period =
2
B
19. Given a modified Sine equation be able to figure out the horizontal shift.
Horizontal shift =
C
.
B
C
is negative (after you have rewritten the equation in "standard form")
B
C
then the horizontal shift is
to the left.
B
Note if
20. Given the graph of a modified Sine function be able to select the corresponding equation.
21. Given a modified Sine equation be able to pick out the correct graph.
22. Make sure that you know the domains of Sine, Cosine and Tangent.
Domain of both the sine and cosine function is  ,  
Domain of the tangent function is all real numbers except
k
where k is any odd integer.
2
Remember that the tangent function has vertical asymptotes at any location where the cosine function = 0
23. Given two sides of a triangle and the included angle, be able to estimate the area.
Area 
1
1
1
ab sin C  bc sin A  ac sin B Also remember your decimal approximations for
2
2
2
3
3
2
2
24. Given two angles and one side of an angle, be able to find one of the remaining sides.
Law of sines (unless it is a SAS in which case use law of cosines).
b
a
a

 b  sin B 
sin B sin A
sin A
24
25. If cos 2 A   . What is the value of sin A ? Make sure that YOU know all three versions of identity #21!
25
24
24
cos 2 A  1  2sin 2 A  
 1  2sin 2 A   1 
 2sin 2 A
25
25
25 24
49
49
 
 2sin 2 A  
 2sin 2 A 
 sin 2 A
25 25
25
50
sin A  
49
7

50
5 2
2
2
26. Be able to use the Law of Sines to solve problems like. In triangle ABC,
B  600
c  8 and
b
c

sin B sin C
sin C 
 b  sin B 
1
4
Find the length of side b
c
8
3
 sin 600  
16  8 3
1
sin C
2
4
27. Be able to solve a SSS triangle for the largest angle. Since you are not allowed a calculator then obviously the side
lengths must be chosen in such a way to yield an angle that you can obtain WITHOUT the use of a calculator!
28. Be able to solve a SAS triangle for the missing side. You should be able to estimate the answer without a calculator
given your knowledge of the few basic decimals I told you to know.
3  1.732
3
 .866
2
2  1.414
2
 .707
2
29. Be able to find an exact value for a trig expression involving lesser used trig functions. Example:
csc600  cot 300  csc 450
3
1
cos 300
1
1
1
2
csc 60  cot 30  csc 45 



 2 

 3 2
0
0
0
1
sin 60
sin 30
sin 45
3
2
3
2
2
2
2 3
2 3 3 3 3 2  3 3 2
3 2 3

 3 2 

or
3
3
3
3
0
0
0
30. Be able to compute compositions of regular trig functions with inverse trig functions. Like #7 BUT different trig and
inverse trig functions AND you will need to draw a triangle rather than use the unit circle!
31. Given an angle in standard position and the coordinates of a point on the terminal side be able to obtain the value
of any trig function for that angle.
32. Can you find the exact value of sin 3150  tan1350 ?
sin 3150  tan1350  
33. If cos  
3
4
2
2
2 2  2 2
 1  
1  
 
or
2
2
2 2
2
2 2
2
and  is in quadrant IV, find the value of sin2 .
Draw a right triangle in quadrant IV with  in standard position. The reference angle for  will be one of the angles in
your right triangle. Label the side adjacent to  (the side on the positive x-axis) 3 and label the hypotenuse 4. Use the
Pythagorean theorem to find the y value and obtain  7 . Then use your trig identity for sin2 .

7  3 
3 7
sin2  2sin  cos   2  
    
8
 4  4 
34. Given a modified cosine equation, be able to find the period.
35. What is cos7 x cos3x  sin 7 x sin 3x equivalent to? Be sure you know the sum and difference identities for both
sine and cosine.
cos 7 x cos3x  sin 7 x sin 3x  cos  7 x  3x   cos 10 x 
36. Be able to simplify a trig expression involving cos 2
Remember cos 2  cos2   sin 2   2cos2   1  1  2sin 2 
37. Make sure that you know identity #9 and the several other versions of it that we have used in this course.
cos2   sin 2   1
cos2   1  sin 2 
sin 2   1  cos2 
38. Make sure that you understand the relationship between a trigonometric function and its inverse.
If
cos A  x then cos1 x  A
39. Be able to find a reference angle for a given degree measure.


40. Can you find an angle x in the third quadrant where cot x  300  tan x ? What about in the other quadrants?
Go look at your unit circle! Remember that cotangent is x/y and tangent is y/x. Can you find an angle whose tangent
matches the cotangent of an angle 30 degrees less? (Answer x  1500 ). What about the other quadrants?
41. Can you find an angle coterminal to another angle?
42. Be sure to know for what values sin,sin 1 ,cos,cos 1 , tan
and
tan 1 are defined. i.e. What are the domains of
each?
sine and cosine both have domain  ,   .
tangent has domain all real numbers except
sin 1 and cos 1 both have domain  1,1
k
where k is any odd integer.
2
tan 1 has domain  ,  
43. Be able to find a solution to a trig equation involving sine and cosine.
1
1
44. Can you find cot 1 ? sec  2  ?

1
cot 1 1  tan 1    tan 1 (1) 
4
1
 1  2
sec1  2   cos 1    
 2 3
45. Be able to solve a very basic sine or cosine equation in the interval 00 ,3600
46. Can you simplify basic expressions like
cot x
?
cos x
cos x
cot x sin x cos x 1
1




 csc x
cos
x
cos x
sin x cos x sin x
1

47. Be sure that you know the “word” definitions for each of the six regular trig functions. i.e. sin  
48. Be sure that you know the range of each of arcsin x, arc cos x
  
  2 , 2 
0,  
and
opposite
hypotenuse
arctan x
  
  ,  respectively!
 2 2
49. Know when you should use each of the following to find the missing side of a triangle.. Pythagorean theorem, Law
of Sines, Law of Cosines, Similar triangles.
50. Be able to calculate the area of a triangle when you are given two sides and the included angle.
Area 
1
1
1
ab sin C  bc sin A  ac sin B
2
2
2