Mathematical Investigations III
Name
Mathematical Investigations III
Trigonometry - Modeling the Seas
REFERENCE TRIANGLES
We began this study of trigonometry by using a unit circle and string to create the sine
and cosine curves. Here, we return to the unit circle. Recall that we measure angles in radians
starting along the positive x-axis as shown. Measures of 0, , and 2 radians are shown. Why
are 0 and 2 radians shown at the same location on the circle?
1.
First consider the unit circle along with the graph of y = sin x. For each point, where
A =π/4, but B and C are not specifically known, indicate all possible points on the graph of
the function that correspond.
A /4
B

1
0
2
C
2.
The 30-60-90 triangle and the 45-45-90 triangle are important. If the hypotenuse has
length 1 in each triangle below, use geometry to fill in the lengths of the remaining sides.
1
1
30o
3.
45o
These few values help us to find the values of the sine and cosine of the special angles.
First, let's do some "translation." For each degree measure, find the corresponding radian
measure, and vice versa.
30° 
45° 
120° 
225° 
180° 
210° 
/2 
/3 
5/6 
3/4 
3/2 
4/3 
Trig. 5.1
Rev. S05
Mathematical Investigations III
Name
4.
For each angle, locate it on the unit circle by drawing the appropriate radius and complete
the right triangle by drawing a vertical segment to the horizontal axis. This triangle is called
a reference triangle. Use the lengths of the sides from the preceding triangles to find the
sine and cosine of each angle.
a.
  /4
cos( ) =
sin( ) 
b.
  /3
cos( ) =
sin( ) 
c.
  /6
cos( ) =
sin( ) 
d.
  5 / 6
cos( ) =
sin( ) 
e.
  4 / 3
cos( ) =
sin( ) 
f.
  7 / 4
cos( ) =
sin( ) 
g.
  5 / 3
cos( ) =
sin( ) 
Trig. 5.2
Rev. S05