7-5 The Four Other Trig Functions

The following are the defintions of the other 4 trig functions
tangent of q:  tan q = y/x
cotangent of  q:  cot q = x/y
secant of q: sec q = r/x
cosecant of q : csc q = r/y

These four trig functions can be written in terms of sin and cos of q
The last one shows that the cotangent and tangent are reciprocal functions.  Secant and cosine, as well as Cosecant and sine are reciprocal funtions.

It is easy to memorize the signs of the six trig functions.  All are positive in Quad I, Sine and Csc are positive in quad II, Tan and Cot are positive in Quad III, while Cos and Sec are positive in quad IV.

Graphs of the other trig functions:
Tangent graph:
Demonstration of the Tangent Graph (Manipula Math)
Period length is p
zeros at 0, p, 2p
undefined at p/ 2, 3p / 2
This corresponds to the zeros of sin -- this is where the tangent crosses the x-axis, and
to the zeros of the cos -- this is where the tangent is undefined.

Cotangent graph:
Period length is p
zeros are at: p/ 2, 3p / 2
undefined at: 0, p, 2p
This again corresponds with the zeros of the sine and cosine, simply reversed from the tangent graph.

Secant graph
The blue graph is the secant graph.  We can generate the secant graph by knowing the graph of the cosine.  Remember that they are reciprocal functions.  When the cosine is zero, the secant is undefined.  When the cosine is at a maximum value, the secant is a minimum.  When the cosine is at a minimum, the secant is a maximum.
Period length is 2p
Keep in mind that when a graph is undefined, there is a vertical asymptote.

Cosecant Graph:
The blue graph is the cosecant graph.  This graph has the same relationship to the sine graph that the cosine and secant graph had.
Period length is 2p

Example problems
1)  Find the other trig functions if sin q = 3/5 and q is in quadrant II.
        y = 3, r = 5, therefore x = -4.  Negative because we are in quad II
                cos q = -4/5
                tan q = -3/4
                csc q =  5/3
                sec q = -5/4
                cot  q = -4/3

2)  Find each of the values for the trig functions using your Ti-82 graphing calculator.  Round to 4 significant digits.

        a)  Tan 115o
                    Make sure you are in degree mode.  Type
                    tan 115.  Answer  -2.145
        b)  Cot 95o
                    Since cot and tan are reciprocals and you don't have a cot button, type it in as:
                    1/tan 95     ----->  Answer  -.0875
        c)  Csc 5
                    Make sure you are in radian mode.
                    since we don't have a csc button but we remember that csc is the reciprocal of sin, type it in as:
                    1/sin 5  ----------->  Answer:  -1.043
        d)  Sec 11
                    Since sec and cos are reciprocals, type as:
                    1/cos 11  --------->   Answer:  226.0

That's it for this section.  Let's head into the last section on inverses.
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