Review of Trigonometric Functions

Solve , where .

Solution Using the double-angle identity , you can rewrite the
equation as follows .
 

	Given equation
	Trigonometric identity
	Quadratic form
	Factor .

 If then and or if
then and .Thus, for there are three solutions .

,or

Graphs of Trigonometric Functions

A function f is periodic if there exists a nonzero number p such that
for all x in the domain of f. The smallest such positive value of p (if it exists) is
the period of f. The sine, cosine, secant, and cosecant functions each have a period
of  2π and the other two trigonometric functions have a period of π as shown in
Figure D.37.

The graphs of the six trigonometric functions
Figure D.37

Note in Figure D.37 that the maximum value of sinx and cosx is 1 and the
minimum value is -1 .The graphs of the functions and
oscillate between -a and a, and hence have an amplitude of Furthermore,
because b x =0 when x= 0 and when ,it follows that the functions
and each have a period of .The table below
summarizes the amplitudes and periods for some types of trigonometric functions.

Function	Period	Amplitude
y = a sin bx or y = a cos bx
y = a tan bx or  y = a cot bx		Not applicable
y = a sec bx or  y = a csc bx		Not applicable

Figure D.38

Sketch the graph of .

Solution The graph of has an amplitude of 3 and a period of
. Using the basic shape of the graph of the cosine function, sketch one
period of the function on the interval ,using the following pattern.

Maximum:

Minimum:

Maximum:

By continuing this pattern, you can sketch several cycles of the graph, as shown in
Figure D.38.

Horizontal shifts, vertical shifts, and reflections can be applied to the graphs of

trigonometric functions, as illustrated in Example 7.

Solution

a. To sketch the graph of shift the graph of  to the left
π/2 units, as shown in Figure D.39(a).

b. To sketch the graph of shift the graph of up two units ,
as shown in Figure D.39(b).

c. To sketch the graph of shift the graph of up
two units and to the right π/4 units, as shown in Figure D.39(c).

(a) Horizontal shift to the left	(b) Vertical shift upward	(c) Horizontal and vertical shifts

Transformations of the graph of
Figure D.39