Honors Geometry
Advanced Trigonometry
4. Trigonometric Functions with Exact Values
For most angles, the values of the sine,
cosine, and tangent functions are nonrepeating decimals which must be
approximated by rounding.
For example, sin 40 = 0.642787609…
This is usually rounded to 0.6428 in
places such as the trig tables in the back
of your Geometry book.
ordinate represents the sine. For
example, if an angle on the unit circle
terminates at (-0.6, 0.8), its cosine is
-0.6, and its sin is 0.8. The tangent is
still y/x, so it would be 0.8/-0.6, or -4/3.
However, some angles have exact
values. All the quadrant angles (0, 90,
180, 270) and the special angles (30,
45, 60) have exact values for their
sines, cosines, and tangents.
Naturally, any angle which is coterminal with the angles mentioned
above also has exact values for trig
functions. For example, 450 and -180
have trig functions with exact values
because they are co-terminal with 90
and 180.
For example, a 180 angle terminates at
(-1,0). So its cosine is -1, its sine is 0,
and its tangent (0/-1) is 0.
In addition, any angle for which a 30,
45, 60 angle is the reference angle has
a sine, cosine, and tangent with exact
values. For example, 135 has exact trig
values because its reference angle is the
special angle 45.
We can use any size circle we wish in
trigonometry, but the most convenient
circle is the unit circle, which has a
radius of one unit and a center at the
origin.
y y
 y
r 1
x x
 x
Cos θ =
r 1
Sin θ =
So, for any point on the unit circle, the x
co-ordinate represents the cosine of the
angle terminating there, and the y-co-
Here you can see that the sine of 30 is
3
1
, the cosine is
, and the tangent
2
2
3
3
3
1
1
is
. ( 
=
=
)
2
3
2
3
3
Honors Geometry
Advanced Trigonometry
4. Trigonometric Functions with Exact Values
Any angle whose measure is evenly
divisible by 15 has exact values for sine,
cosine, and tangent.
Exercises
Fill in – then memorize – the table.
θ
sine
cosine tangent
0°
90°
The circle above shows that the sine and
2
cosine of 45 each equal
. The
2
tangent of 45 equals 1 (a very useful
fact down the road).
180°
270°
You could continue constructing
triangles to show that sin 60 =
cos 60 = ½ , and tan 60 =
3
,
2
30°
3.
45°
Sine, cosine, and tangent of other
angles
Sin 630
By co-terminal angles,
sin 630 = sin 270 = -1
Cos 240
The reference angle for 240 is 60.
Cosine is negative in Quad III, so
cos 240 = -cos 60 = - ½
Tan -225
By co-terminal angles,
tan(-225) = tan 135
The reference angle for 135 is 45.
Tangent is negative in Quad II, so
tan(-225) = tan 135 = -tan 45 = -1
60°
Give exact values for the quantity.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
sin(-90)
cos 900
tan(-720)
sin 810
cos(-270)
tan 450
sin 135
cos 120
tan 330
sin 210
cos 315
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
tan 150
sin 150
cos 300
tan 210
sin 120
cos 135
tan 225
sin 660
cos(-45)
tan -315