May 9, 2016
Dear Future Honors Pre-Calculus Students,
Attached you will find the summer packet for Honors PreCalculus. It is due the first day of the school year and will be your
first grade of the year. This packet is not long and is not intended
to keep you busy this summer. We simply want your
Trigonometry skills to be fresh when we start the year. It would
benefit you to wait and work on it a week or two prior to the year
to help refresh your trig skills.
In other words, don’t complete it now because that will defeat the
purpose.
We look forward to working with you next year. Have a great
summer and we’ll see you in August.
Sincerely,
The Math Department
UNIT CIRCLE
What you see above is a unit circle. A unit circle has a radius of one (one unit). Around the unit circle are
degree values of the special angles starting with 0 on the positive side of the x-axis (initial side) and
moving counter-clockwise around the circle to the terminal side (where the angle ends). For instance the
angle 210 starts at 0 , moves counter-clockwise through the first and second quadrants then terminates
30 into the third quadrant. A reference angle is the acute angle between the nearest side of the x-axis and
the terminal side of the angle. A special angle as mentioned above is an angle whose reference angle is
30 , 45 , or 60 . In addition, there are quadrantal angles whose terminal side lie on either the xaxis or y-axis ( 90 ,180 , 270 ,360 , etc. ).
either
Also included in the above unit circle picture are the corresponding radian values for the special angles.
57 and  radians is 180 . That makes it easy to work with these values as



well because 30  , 45  and 60  . One easy way to remember the radian values is to think
6
4
3
   7
of what times 30,45 or 60 equals the angle that you want. For instance, 210  7  30  7   
so
6 6
7
5
   5
210 
so 300 
. One additional one is 300  5  60  5   
.
6
3
3 3
One radian is approximately
Converting between Radians and Degrees
Converting angles is quite easy. For the special angles, I would use the method that I
mentioned above. However, if it is not one of the special angles then you have to
convert by multiplying by a specific ratio.
To convert a degree value to a radian value you multiply by
value to a degree value
180

. This works because

180
and from a radian
180
180
and
= 1.
For example, let’s convert 257 to radians. We take 257 
7
to degrees.
8
Now let’s convert



180

257
.
180
7 180

 157.5
8 
Evaluating Trig. Functions
Trig. Functions
sin
 = opposite = y
hypotenuse
cos
y
 = hypotenuse = r
opposite
sec
x
 = adjacent = x
opposite
csc
r
 = opposite = y
adjacent
cot
r
 = adjacent = x
hypotenuse
tan
Common Triangles used in Trig.
y
 = hypotenuse = r
adjacent
x
There are many ways to evaluate trig functions. We are going to focus on one method.
However, feel free to use whatever works for you.
First is to know the unit circle and the (x,y) values at each special angle around the edge
of the circle. These values come from the special right triangles from the last page.
Since the radius of the circle is 1, it makes evaluating trig functions really easy! The xvalue is the cosine value while the y-value is the sine value.
 4
For example let’s solve sin 
 3

.

y
4
Well sin   and at
we have
r
3
 4
sin 
 3
 1
3
y
  , 
 Now take 
2 
r
 2
 3
2   3 so
1
2
  3
.

2

Let’s try one more. Cot 150  . OK, Cot 
x
So we take 
y
 3 1
x
,  .
and the point at 150 is 
2
2
y

 3
2   3 . So the Cot 150   3
 
1
2
Some students think you must memorize the points all the way around but that is not the
case. Just remember the 1st quadrant and those values repeat for every quadrant at the
corresponding reference angles. All you really do is change the signs and that is simple!
Exercises
What quadrants are the following angles in and find the reference angle. If the angle is
given in degrees then find the reference angle in degrees. If the angle is given in radians,
then find the reference angle in radians.
1)
120
2)
75
3)
5
4
4)
11
6
5)
330
6)
150
7)

4
8)
2
3
9)
7
6
10)
180
Convert the following angles to radians.
11)
135
12)
210
13)
30
14)
300
15)
270
16)
120
17)
225
18)
180
19)
60
20)
240
Convert the following angles to degrees.
21)

4
22)

2
23)
4
3
24)
7
6
25)
7
4
26)
2
3
27)
3
4
28)
2
29)

6
30)
11
3
Evaluate the trig functions
31)
sin(60 ) =
32)
cos (135
)=
33)
tan(180
34)
cos(240
)=
35)
sec (150
36)
csc(45
37)
cot(210
38)
sin(270
39)
 7 
tan 
 =
 4 
40)
 2 
cos 
 =
 3 
41)
sec( 2 ) =
42)
 11 
sin 
 =
 6 
43)
 
csc   =
4
44)
 4 
cos 
 =
 3 
)=
)=
)=
)=
)=
Find the first quadrant angle (  0    90

1
Ex. sin 1   = 30 or
6
2
 for each inverse trig function.
this can also be written as

1
arcsin   = 30 or
6
2
45)
cos 1
 3

 =
2


46)
tan 1 (0) =
47)
 3
arcsin 
 =
 2 
48)
cot 1 1 =
49)
 3
arctan 
 =
 3 
50)
 2
sin 1 
 =
 2 
51)
cot 1 (0) =
52)
 2
cos 1 
 =
 2 
53)
arctan( 3 )=
54)
arcsin (0) =