Download Pre-Class Problems 2 for Thursday, September 11 These are the

Pre-Class Problems 2 for Thursday, September 11
These are the type of problems that you will be working on in class. These
problems are from Lesson 2 and Lesson 3.
Solution to Problems on the Pre-Exam.
You can go to the solution for each problem by clicking on the problem letter.
1.
Find the exact value of the six trigonometric functions (cosine, sine, tangent,
secant, cosecant, and cotangent) of the following angles.
NOTE: The objective of these problems is to use Unit Circle Trigonometry
to find the value of any of the six trigonometry functions of the angles whose
terminal sides are located on one of the coordinate axes.
2.
3
2
a.
 0
b.
  180 
c.
 
d.
  270 
e.
  90 
f.
 
g.
 

2
h.
   360 
i.
  2
Find the exact value of the six trigonometric functions (cosine, sine, tangent,
secant, cosecant, and cotangent) of the following angles.
NOTE: The objective of these problems is to use either Unit Circle
Trigonometry or Right Triangle Trigonometry to find the value of any of the
six trigonometry functions of these three acute angles.
a.
3.
  30 
b.
 

3
c.
 

4
For each angle, determine the location of the angle and then find the
reference angle of the angle if it has one. Show your work for finding the
reference angle.
NOTE: The objective of these problems is to learn how to find the reference
angle of an angle if it has one. The reference angle of an angle is an acute
angle.
a.
  210 
d.
 
g.
 
j.
   270 
b.
   325 
2
3
e.
 
3
8
h.
   135 
k.
 
6
17
45 
26
c.
 
f.
 
i.
 
15 
11

2
Additional problems available in the textbook: Page 137 … 9, 10, 13, 14, 21, 22,
28, 29, 30. Page 134 … Examples 1a and 1b. Page 140 … Examples 2 and 3.
Page 157 … 45 - 50. Page 152 … Example 4.
Solutions:
1a.
 0
Animation of the making of the 0 angle.
NOTE: The angle of 0 radians is the same as the angle of 0  . The terminal
side of this angle is the positive x-axis.
NOTE: Unit Circle Trigonometry: P (  )  ( cos  , sin  )
P ( 0 )  P ( 0  )  ( 1, 0 )
cos 0  x  1
sec 0 
1
1
 1
x
1
sin 0  y  0
csc 0 
1
1

 undefined
y
0
tan 0 
y
0

 0
x
1
cot 0 
x
1

 undefined
y
0
Back to Problem 1.
1b.
  180 
Animation of the making of the 180  angle.
NOTE: The angle of 180  is the same as the angle of  radians. The
terminal side of this angle is the negative x-axis.
NOTE: Unit Circle Trigonometry: P (  )  ( cos  , sin  )
P (180  )  P (  )  (  1, 0 )
cos 180   x   1
sec 180  
1
1

 1
x
1
sin 180   y  0
csc 180  
1
1

 undefined
y
0
cot 180  
x
1

 undefined
y
0
tan 180  
y
0

 0
x
1
Back to Problem 1.
1c.
 
3
2
Animation of the making of the 
3
angle.
2
3
radians is the same as the angle of  270  . The
2
terminal side of this angle is the positive y-axis.
NOTE: The angle of 
NOTE: Unit Circle Trigonometry: P (  )  ( cos  , sin  )
 3 
P 
  P (  270  )  ( 0 , 1)
 2 
 3 
cos  
  x  0
2 

1
1
 3 
sec  

 undefined
 
x
0
 2 
 3 
sin  
  y 1
 2 
1
1
 3 
csc  
 1
 
y
1
 2 
y
1
 3 
tan  

 undefined
 
x
0
 2 
x
0
 3 
cot  

 0
 
y
1
 2 
Back to Problem 1.
1d.
  270 
Animation of the making of the 270  angle.
NOTE: The angle of 270  is the same as the angle of
3
radians. The
2
terminal side of this angle is the negative y-axis.
NOTE: Unit Circle Trigonometry: P (  )  ( cos  , sin  )
P ( 270  )  P (  )  ( 0 ,  1)
cos 270   x  0
sin 270   y   1
tan 270  
y
1

 undefined
x
0
sec 270  
1
1

 undefined
x
0
csc 270  
1
1

 1
y
1
cot 270  
x
0

 0
y
1
Back to Problem 1.
1e.
  90 
Animation of the making of the 90  angle.
NOTE: The angle of 90  is the same as the angle of

radians. The terminal
2
side of this angle is the positive y-axis.
NOTE: Unit Circle Trigonometry: P (  )  ( cos  , sin  )
 
P ( 90  )  P    ( 0 , 1 )
 2
cos 90   x  0
sec 90  
1
1

 undefined
x
0
sin 90   y  1
csc 90  
1
1
 1
y
1
cot 90  
x
0

 0
y
1
tan 90  
y
1

 undefined
x
0
Back to Problem 1.
1f.
 
Animation of the making of the   angle.
NOTE: The angle of   radians is the same as the angle of  180  . The
terminal side of this angle is the negative x-axis.
NOTE: Unit Circle Trigonometry: P (  )  ( cos  , sin  )
P (   )  P (  180  )  (  1, 0 )
cos (   )  x   1
sec (   ) 
1
1

 1
x
1
sin (   )  y  0
csc (   ) 
1
1

 undefined
y
0
tan (   ) 
y
0

 0
x
1
cot (   ) 
x
1

 undefined
y
0
Back to Problem 1.
1g.
 

2
NOTE: The angle of 
Animation of the making of the 

angle.
2

radians is the same as the angle of  90  . The
2
terminal side of this angle is the negative y-axis.
NOTE: Unit Circle Trigonometry: P (  )  ( cos  , sin  )
  
P     P (  90  )  ( 0 ,  1 )
 2
  
cos     x  0
 2
1
1
  
sec    

 undefined
x
0
 2
  
sin     y   1
 2
y
1
  
tan    

 undefined
2
x
0


Back to Problem 1.
1
1
  
csc    

 1
y
1
 2
x
0
  
cot    

 0
2
y

1


1h.
   360 
Animation of the making of the  360  angle.
NOTE: The angle of  360  is the same as the angle of  2  radians. The
terminal side of this angle is the positive x-axis.
NOTE: Unit Circle Trigonometry: P (  )  ( cos  , sin  )
P (  360  )  P (  2  )  ( 1, 0 )
cos (  360  )  x  1
sec (  360  ) 
1
1
 1
x
1
sin (  360  )  y  0
csc (  360  ) 
1
1

 undefined
y
0
cot (  360  ) 
x
1

 undefined
y
0
tan (  360  ) 
y
0

 0
x
1
Back to Problem 1.
1i.
  2
Animation of the making of the 2  angle.
NOTE: The angle of 2  is the same as the angle of 360  . The terminal side
of this angle is the positive x-axis.
NOTE: Unit Circle Trigonometry: P (  )  ( cos  , sin  )
P ( 2  )  P ( 360  )  ( 1, 0 )
cos 2   x  1
sec 2  
1
1
 1
x
1
sin 2   y  0
csc 2  
1
1

 undefined
y
0
tan 2  
y
0

 0
x
1
cot 2  
x
1

 undefined
y
0
Back to Problem 1.
2a.
  30 
Animation of the making of the 30  angle.
NOTE: The angle of 30  is the same as the angle of

radians. The terminal
6
side of this angle is in the I quadrant.
Using Unit Circle Trigonometry: P (  )  ( cos  , sin  )
  
P ( 30  )  P    
 6  
3
2
cos 30   x 
sin 30   y 
tan 30  
y

x
3 1
, 
2
2 
1
2
1
2 
3
2
1
3
Using Right Triangle Trigonometry:
sec 30  
1

x
csc 30  
1
 2
y
cot 30  
x

y
2
3
3
2 
1
2
3
60 
2
1
30 
3
3
2
sec 30  
hyp

adj
opp
1

hyp
2
csc 30  
hyp
2

 2
opp
1
opp

adj
cot 30  
adj

opp
cos 30  
adj

hyp
sin 30  
tan 30  
1
3
2
3
3

1
3
Back to Problem 2.
2b.
 

3

Animation of the making of the angle.
3

NOTE: The angle of
radians is the same as the angle of 60  . The terminal
3
side of this angle is in the I quadrant.
Using Unit Circle Trigonometry: P (  )  ( cos  , sin  )
 1
 
P    P ( 60  )   ,
 2
 3

3
2




cos

1
 x 
3
2
sin

 y 
3
3
2

y


3
x
3
2 
1
2
tan
sec

1

 2
3
x
csc

1
2


3
y
3
cot
3

x


3
y
1
2 
3
2
1
3
Using Right Triangle Trigonometry:
60 
2
1
30 
3
cos

adj
1


3
hyp
2
sec

hyp
2


 2
3
adj
1
sin

opp


3
hyp
3
2
csc

hyp
2


3
opp
3
tan

opp


3
adj
3

1
cot

adj
1


3
opp
3
Back to Problem 2.
3
2c.
 

4
Animation of the making of the
NOTE: The angle of

angle.
4

radians is the same as the angle of 45  . The terminal
4
side of this angle is in the I quadrant.
Using Unit Circle Trigonometry: P (  )  ( cos  , sin  )

 
P    P ( 45  )  

 4

2
,
2
2
2




cos

 x 
4
2
2
sec

1


4
x
2

2
2
sin

 y 
4
2
2
csc

1


4
y
2

2
2

x


4
y
2
2 1
2
2
tan

y


4
x
2
2 1
2
2
cot
Using Right Triangle Trigonometry:
45 
2
1
45 
1
cos

adj


4
hyp
1
2
sec

hyp


4
adj
2

1
2
sin

opp


4
hyp
1
2
csc

hyp


4
opp
2

1
2
tan

opp
1

 1
4
adj
1
cot

adj
1

 1
4
opp
1
Back to Problem 2.
3a.
  210 
Animation of the making of the 210  angle.
210   180  and 210   270    is in III
180    '  210    '  30 
  210 
'
OR  '  210   180   30 
Answer: 30 
Back to Problem 3.
3b.
   325 
Animation of the making of the  325  angle.
325   180  and 325   270    is in I
325    '  360    '  35 
   325 
'
OR  '  360   325   35 
Answer: 35 
Back to Problem 3.
3c.
 
45 
26
Animation of the making of the
45 
angle.
26
45 
26 
45 
3
39 
  


  is in IV
and
26
26
26
2
26
 
45 
7
  '  2   ' 
26
26
45 
7

'

2



OR
26
26
45 
26
'
Answer:
7
26
Back to Problem 3.
3d.
 
2
3
Animation of the making of the
2
angle.
3
2
3
2

1.5 
  


  is in II
and
3
3
3
2
3
2

 '    ' 
3
3
'
 
2
3
OR  '   
Answer:
2


3
3

3
Back to Problem 3.
3e.
 
6
17
Animation of the making of the 
6
angle.
17
6
17 
6

8.5 
  


  is in IV
and
17
17
17
2
17
 
6
17
'
0  ' 
6
6
 ' 
17
17
OR  ' 
6
6
 0 
17
17
Answer:
3f.
 
6
17
Back to Problem 3.
15 
11
Animation of the making of the 
15 
angle.
11
15 
11
15 
3
16.5 
  


  is in II
and
11
11
11
2
11
'
 
  ' 
15 
4
 ' 
11
11
OR  ' 
15 
4
  
11
11
15 
11
Answer:
4
11
Back to Problem 3.
3g.
 
3
8
Animation of the making of the
3
8
3

4
  


  is in I
and
8
8
8
2
8
3
angle.
8

3
8
'
0  ' 
3
3
 ' 
8
8
OR  ' 
3
3
 0 
8
8
Answer:
3
8
Back to Problem 3.
3h.
   135 
Animation of the making of the  135  angle.
135   180  and 135   90    is in III
135    '  180    '  45 
OR  '  180   135   45 
'
   135 
Answer: 45 
Back to Problem 3.
3i.
 

2
Animation of the making of the

angle.
2
Answer: No reference angle
 

2
Back to Problem 3.
3j.
   270 
Animation of the making of the  270  angle.
Answer: No reference angle
   270 
Back to Problem 3.
3k.
 
Animation of the making of the  angle.
Answer: No reference angle
 
Back to Problem 3.
Solution to Problems on the Pre-Exam:
1b.
tan 180 
Back to Page 1.
Animation of the making of the 180  angle.
P (180  )  (  1, 0 )
tan 180  
y
0

 0
x
1
Answer: 0
1a.
Reference angle of  
4
3
Animation of the making of the
4
angle.
3
4
3
4
3
4.5 
  


  is in III
and
3
3
3
2
3
4

 ' 
3
3
4


'




OR
3
3
  ' 
 
4
3
'
Answer:  ' 

3
1b. Reference angle of   180  Animation of the making of the 180  angle.
Answer:  '  No reference angle
  180 
1c.
Reference angle of    315  Animation of the making of the  315  angle.
315   180  and 315   270    is in I
315    '  360    '  45 
   315 
'
OR  '  360   315   45 
Answer:  '  45 
5.
Reference angle of   290 
Animation of the making of the 290  angle.
290   180  and 290   270    is in IV
290    '  360    '  70 
OR  '  360   290   70 
  290 
'
Answer:  '  70 
6.
Reference angle of   
11
14
Animation of the making of the 
11
angle.
14
11
14 
11

7
  


  is in III
and
14
14
14
2
14
11
3
 '    ' 
14
14
OR  '   
'
  
11
14
11
3

14
14
Answer:  ' 
3
14