Unit Circles |
Consider the unit circle given by the equation
x² + y² = 1. The radius is therefore 1. The
equation of the arc length of an angle is s = r
where s is the arc length, r is the radius, and ? is the
central angle. When the radius is 1, the arc length equals
the angle. Using unit circles makes it easier to relate
to real life applications. |
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Trigonometric Functions
|
| Consider the unit circle given by the equation
x² + y² = 1. The radius is therefore 1. The
equation of the arc length of an angle is s = r
where s is the arc length, r is the radius, and
is the central angle. When the radius is 1, the arc length
equals the angle. Using unit circles makes it easier to
relate to real life applications. |
Definitions of Trigonometric Functions:
|
Let t be a real number and let (x, y) be
the point on the unit circle corresponding to the angle
t.
sin t = y
cos t = x
tan t = y/x x 
0
csc t = 1/y y
0
sec t = 1/x x
0
cot t = x/y y
0
Cosecant, secant, and cotangent are the reciprocals of
sine, cosine, and tangent respectively. The tangent and
secant of t are not defined when x = 0. This is shown
in the following examples:
t =  /2 corresponds
to (0, 1) because cos(/2)
= 0 and sin(/2)
= 1, it follows that tan(/2)
and sec(/2)
are undefined. The cotangent and cosecant functions are
also undefined when y = 0.
t = 0 corresponds to (1,0), therefore cot 0 and csc 0
are undefined. |
Domain and Period
|
| The domain of the sine and cosine functions
is the set of real x values that fall on the function.
Remember that r = 1, sin t = y, and cos t = x. As the
radius is 1, |
| |
|
| -1 = y = 1 |
-1 = x = 1 |
Therefore the values of sine and cosine also range
between -1 and 1: |
| |
|
| -1 = sin t = 1 |
-1 = cos t = 1 |
|
Adding 2
to each value of t completes another full revolution around
the unit circle. Therefore, the values of sin(t + 2)
and cos(t + 2)
correspond to those of sin t and cos t. Adding any multiple
of 2p, positive or negative, will yield the same results.
This means that:
sin(t + 2n)
= sin t and cos(t + 2n)
cos t
for any integer n and real number t. Functions that repeat
in cycles like this are called periodic. A function is
periodic if there is a positive number c such that
f(t +c) = f(t)
The smallest number c that makes f periodic is called
the period of f. It is shifted by c so that after the
shift it the same as the original function. |
 |
Even and Odd Trigonometric Functions
|
The cosine and secant functions are even.
cos(-t) = cos t sec(-t) = sec t
The sine, cosecant, tangent, and cotangent functions are
odd.
sin(-t) = -sin t csc(-t) = -csc t
tan(-t) = -tan t cot(-t) = -cot t |
