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The Unit Circle - Simple Trigonometry.
Trigonometry is the study of angles and the physical relationships between angles and geometry. To start,
we use the unit circle, which is a circle of radius 1 unit, centered at the origin.
Part 1: Finding coordinates of points on the unit circle.
Four of the coordinates are easy: (1,0), (0,1), (-1,0) and (0,-1). These lie on the x and y axes and are called
cardinal points (i.e. like the points on a compass). Draw any ray from the origin intersecting the circle.
Call the intersection point P. This ray has an angle of theta "". Therefore, the coordinates of point P are a
function of the angle . We make the following definition:
Given point P on the unit circle. Point P has coordinates (x,y).
x = cos , and y = sin .
Simple geometry (right triangles and pythagoras' formula) will help locate values for some angles such as
30, 45 and 60. Using symmetry on the unit circle you can find values for angles that are multiples of
30, 45 and 60. Your calculator will assist you for other angles.
First quadrant values for cos  and sin . That means 0 <  < 90 (degrees) or 0 <  < /2 (radians).
E
D
C
B
A
Fill in the following table based on the figure above. Give exact values (no decimals).
Angle  in degrees
A. 0
B. 30
Angle  in radians
0
/6
x = cos 
y = sin 
1
0
1/ 2
3/2
C. 45
/4
D. 60
/3
2/2
1/ 2
E. 90
/2
0
2/2
3/2
1
Questions:
1.
What is the distance from point B to the origin? Why?
The distance from point B to the origin is the radius of the circle = 1.
2.
A ray has an angle of  = 20. Find the point's coordinates.
x=cos 20=.9397
3.
y=sin 20=.34202
What would the coordinates of P be for an angle of 390? Why?
Since 390=360+30 an angle of 390 corresponds to a complete revolution plus 30.
Thus the coordinates of P = (cos 390,sin 390)=(cos 30,sin 30)=(
3 1
, )
2 2
Part II: Values of sin  and cos  for the other quadrants.
Using the table in Part I and appropriate symmetric points on the unit circle complete the following table.
Some patterns should be evident. Give exact values. No decimals.
Angle in degrees
120
Angle in radians
2/3
Which quadrant?
2nd
nd
x = cos 
1/ 2
135
3/4
2
150
5/6
2nd
2/2
 3/2
210
7/6
3rd

5/4
3
rd

rd
225
240
4/3
3
300
5/3
4th
315
7/4
4
th
330
11/6
4th
y = sin 
3/2

3/2
2/2
1/ 2
1/ 2
2/2
1/ 2

1/ 2

2/2
3/2
2/2
 3/2
3/2

2/2
1/ 2
Questions:
1.
In quadrant II, the cos  is always ___negative_ and the sin  is always positive .
2.
In quadrant III, the cos  is always ___negative____, and the sin  is always __negative____.
3.
In quadrant IV, the cos  is always __positive___, and the sin  is always ___negative__.
4.
Look at your results for the angles 135, 225 and 315, and compare them to the results for the
angle of 45 from Quadrant I. What do you notice?
The values of sine and cosine are the same in absolute value but the signs are different according
to which quadrant the angle is.
(The angle 45 is called the reference angle for the angles 135, 225 and 315.
5.
What is the reference angle for 120?
The reference angle is 180-120=60
6.
What is the reference angle for 335?
The reference angle is 360-335=25.
7.
In general, if an angle  is in the second quadrant, how would you find its reference angle?
The reference angle is the positive smallest angle formed by the terminal side of  and the xaxis.
If 90 <  <180 the reference angle of  is 180-.
8.
In general, if an angle  is in the third quadrant, how would you find its reference angle?
If 180 < <270 the reference angle of  is  -180.
9.
In general, if an angle  is in the fourth quadrant, how would you find its reference angle?
If 270 <  < 360 the reference angle of  is 360 -  .
Part III: The tangent function (tan ).
The tangent of an angle is equivalent to the slope of the ray. In terms of sin  and cos , we can define
tan  
sin 
. Fill in the following table. Give exact values.
cos 
Angle in degrees
0
30
Tan 
0
45
90
1
undefined
1/ 3
Can you explain why the tangent at 90 is undefined?
The ray at 90 is vertical, thus the slope is undefined.
Part IV: General Practice
1.
An angle of  = 2/3 is given.
a)
What is this angle in degrees?
To convert to degrees we multiply the radian measure by 180/. Thus the degree
measure is 2(180)/3=60.
b)
Which quadrant is this angle in?
The angle is in the second quadrant.
c)
Give the values for sin , cos  and tan .
sin =
2.
3 / 2 , cos =  1/ 2 , tan  
sin 
 3 .
cos
You are told that sin  = -1/3.
a)
In which two quadrants could  be located?
 could be either in the 3rd or 4th quadrant.
You are also told that cos  is positive. Which quadrant must  be in?
b)
 must be in the 4th quadrant.
Find cos  and tan .
c)
To find cos  we use the Fundamental Identity: sin
2
  cos 2   1 . Solving for cos
cos    1  sin  . Since we know from part b) that
2
2
gives
cos is positive we have,
1
1
8 2 2
cos  1  ( ) 2  1  

3
9
9
3
sin 
 1/ 3
1
tan  


cos 2 2 / 3
2 2
NOTE: In order to correctly solve the following problem remember that, if P is a point on a circle of
radius r determined by an angle  then the coordinates of P are given by
x=r cos , y=r sin .
Also, recall the formula that relates arclength to radian measure of an angle: s=r  where s = arclenght, r =
radius and  = angle in radians.
3.
A circle is centered at the origin with radius 2.
a)
A ray of angle 120 degrees is drawn, intersecting the circle at point Q.
coordinates.
Find Q's
1
x  2 cos120   2( )  1
2
3
y  2 sin 120   2( )  3
2
The coordinates of Q are (1, 3 ) .
b)
If you are at the point (2,0) and travel a distance of 3 units around the full circle travelling
counterclockwise, what will your coordinates be?
First we need to find the radian measure of the angle spanned. Using the formula
s  r with s  3 and r  2 we get   3 / 2 radians.
Thus the coordinates are x  2 cos(3 / 2)  .14147 and y  2 sin( 3 / 2)  1.995 .