Recall that a real number can be interpreted as
the measure of the angle constructed as follows: wrap a piece of
string of length units around the unit circle
(counterclockwise if , clockwise if ) with
initial point *P*(1,0) and terminal point *Q*(*x*,*y*). This gives rise to
the central angle with vertex *O*(0,0) and sides through the points
*P*
and *Q*.
All six trigonometric functions of are defined in terms of
the coordinates of the point *Q*(*x*,*y*), as follows:

Since *Q*(*x*,*y*) is a point on the unit circle, we know that
. This fact and the definitions of the trigonometric
functions give rise to the following fundamental identities:

This modern notation for trigonometric functions is due to L. Euler (1748).

More generally, if *Q*(*x*,*y*) is the point where the circle
of radius *R* is intersected by the angle , then
it follows (from similar triangles) that

**Periodic Functions**

If an angle corresponds to a point *Q*(*x*,*y*) on the unit
circle, it is not hard to see that the angle corresponds
to the same point *Q*(*x*,*y*), and hence that

Moreover, is the smallest positive angle for which Equations 1 are true for any angle . In general, we have for all angles :

We call the number the *period* of the trigonometric
functions and , and refer to these functions as being
*periodic*. Both and are periodic functions as well,
with period , while and are periodic with period
.

**EXAMPLE 1 **
Find the period of the function .

*Solution: *
The function runs through a full cycle
when the angle 3*x* runs from 0 to , or equivalently when *x*
goes from 0 to . The period of *f*(*x*) is then .

**EXERCISE 1 **
Find the period of the function .

Consider the triangle with sides of length and hypotenuse
*c*>0 as in Figure 1 below:

Figure 1 |

For the angle pictured in the figure, we see that

There are a few angles for which all trigonometric functions may be found using the triangles shown in the following Figure 2.

Figure 2 |

This list may be extended with the use of *reference angles* (see
Example 2 below).

**EXAMPLE 1:**
Find the values of all trigonometric functions of the angle
.

*Solution: *
From Figure 2, we see that the angle of
corresponds to the point on the unit circle, and
so

**EXAMPLE 2:**
Find the values of all trigonometric functions of the angle
.

*Solution:*
Observe that an angle of is equivalent to 8
whole revolutions (a total of ) plus , Hence the
angles and intersect the unit circle at the
same point *Q*(*x*,*y*), and so their trigonometric functions are the
same. Furthermore, the angle of makes an angle of
with respect to the x-axis (in the second quadrant). From this we can
see that and hence that

We call the auxiliary angle of the *reference angle* of
.

**
EXAMPLE 3
**
Find all trigonometric functions of an angle in the third
quadrant for which .

Solution: We first construct a point *R*(*x*,*y*) on the terminal side of
the angle , in the third quadrant. If *R*(*x*,*y*) is such a point,
then and we see that we may take *x*=-5 and
*R*=6. Since we find that
(the negative signs on *x* and *y* are
taken so that *R*(*x*,*y*) is a point on the third quadrant, see
Figure 3).

Figure 3 |

It follows that

Here are some Exercises on the evaluation of trigonometric functions.

**EXERCISE 2 **

**(a)**-
Evaluate
(give the
**exact**answer).

**(b)**-
If and , find
(give the
**exact**answer).

**EXERCISE 3 **
From a 200-foot observation tower on the beach, a man
sights a whale in difficulty. The angle of depression of the whale is
. How far is the whale from the shoreline?

**
**

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Wed Dec 4 18:30:59 MST 1996

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