:Basic Trigonometric Values
Some basic values of each trigonometric function can be found by analyzing the symmetries present in the unit circle for various special arc lengths. These arc lengths are equivalent to degree measures of 0°, 30°, 45°, 60°, and 90°, and their multiples.
Trig Values of Multiples of
:
| θ | sin θ | cos θ | tan θ | cot θ | sec θ | csc θ |
| 0 | 0 | 1 | 0 | undefined | 0 | undefined |
|
|
1 | 0 | undefined | 0 | undefined | 0 |
| π | 0 | –1 | 0 | undefined | –1 | undefined |
 |
–1 | 0 | undefined | 0 | undefined | –1 |
Proof: Since the circumference of a circle is given by
the formula 
,
the circumference of the unit circle is 2π. One-fourth of this
distance,
,
will be the arc length of the circle between adjacent coordinate axes.
Therefore, the point with coordinates
will correspond to the arc length
, and the Unit Circle Definition will provide the values of the six
trigonometric functions of
.
The proofs for the other arc lengths are similar.♦
Stated in terms of degrees, the preceding results would be the trigonometric values for 0°, 90°, 180°, and 270°. Because none of these values is an acute angle, there is no proof involving triangles for these values.
Trig Values of Multiples of
:
| θ | sin θ | cos θ | tan θ | cot θ | sec θ | csc θ |
|
|
 |
|
1 | 1 |
 |
|
Proof: Since the line 
bisects the first quadrant, it intersects the unit circle at an arc length
of
from the point
.
Substituting this equation into the equation of the unit circle, we have
,
which can be solved to obtain
.
From the substitution, the variable y has the same value. The
ratio and
reciprocal identities can then be used to produce the other four trig values.♦
Alternate Proof: Since an unit circle arc length of
occurs with a 45° angle, we consider triangle right triangle ABC, with
, and
. Suppose side
AC has length 1. Then side BC also has length 1, and by
the Pythagorean Theorem, side AC has length
. Values for the six trigonometric functions follow by the
Triangle Ratios Theorem.♦
Trig Values of Multiples of
:
| θ | sin θ | cos θ | tan θ | cot θ | sec θ | csc θ |
|
|
 |
 |
 |
 |
 |
2 |
 |
|
|
|
|
2 | |
Proof: Let A be the point
on the unit circle,
and choose points B and C in the first quadrant so that arc length
AB is
, and arc length
AC is
. Assume the
coordinates of B are
, then by symmetry
across the line
, the coordinates of
C are 
. Also,
the line segments AB and BC are equal. Applying the distance
formula, we obtain the equation
. Since point
B lies on the unit circle, we also have
. Solving this
system gives 
as the
coordinates of point B. The definition and
basic identities then
produce the values of the six trigonometric functions.♦
Alternate Proof: Let ABC be an equilateral
triangle. Let point D be the intersection of side BC with the
altitude from vertex A. By the Hypotenuse-Leg Theorem of geometry,
triangles ABD and ACD are congruent. Therefore the altitude
from A bisected side BC. If side AB has length 2,
then BD has length 1, and by the Pythagorean Theorem, side AD has
length . The six trigonometric functions will follow
by the Triangle Ratios Theorem.♦
In addition to these basic values, we can also determine the exact values of the sine function at angles that are multiples of 15°, and multiples of 18°.