Download 7.6 graphs of sine and cosine functions

7.5 Properties of Trigonometric Functions: Unit Circle Approach
The unit circle is a circle of radius r = 1. Let t be a real number and let P = (a, b) be the point on
the unit circle that corresponds to t. Then sin t = b, cos t = a, tan t = b/a, csc t = 1/b, sec t = 1/a, and
cot t = a/b.
The point P on the unit circle that corresponds to t is given. Find the six trig functions:
P=(a, b)
Sin t
Cos t
Tan t
Csc t
Sec t
Cot t

1.
3 1
, 

2
 2

2 14 
,
 

4
4 

The point P on the circle x2 + y2 = r2 that is also on the terminal side of  in standard position is
given. Find the six trig functions:
2.
3.
P=(a, b)
(4, -3)
4.
(-5, 2)
r
Sin t
Cos t
Tan t
Csc t
Sec t
Cot t
PERIOD OF TRIGONOMETRIC FUNCTIONS
The trigonometric functions are periodic because, if we add an integral multiple of  or 2 to , the
value of the function is unchanged. All the functions have a period of 2 except the tangent and
cotangent, which have a period of . Thus, sin ( + 2) = sin , tan ( + ) = tan , etc.
Use this fact to find the exact value of each expression:
5.
cos 390
6.
cot 405
7.
sec 7
3
8.
tan 17
4
9. If cos  = 0.3, find the value of: cos  + cos ( + 2) + cos ( + 8)
DOMAIN AND RANGE
Function
Symbol
Domain
Sine
f() = sin 
(-, )
Cosine
f() = cos 
(-, )
Tangent
f() = tan 
{xx  odd multiples of /2 (90)
Cosecant
f() = csc 
{xx  integral multiples of  (180)
Secant
f() = sec 
{xx  odd multiples of /2 (90)
Cotangent
f() = cot 
{xx  integral multiples of  (180)
11. For what numbers  is the secant not defined?
12. What values are not included in the range of the cosecant function?
6/29/2017
Range
[-1, 1]
[-1, 1]
(-, )
(-, -1]  [1, )
(-, -1]  [1, )
(-, )
7.6 GRAPHS OF SINE AND COSINE FUNCTIONS
Function
Symbol
Domain
Range
Period
Sine
y = f(x) = sin x
[-1, 1]
(-, )
2
The sine function, like all the trig functions except cosine and secant, are odd functions, thus
symmetric to the origin and sin (-) = -sin . It has x-intercepts at integral multiples of . Label xaxis between 2 and y-axis between 1; adjust for transformations. Construct table of values:
1
Sin x
0

/2
3/2
2
-/2
-
-3/2
-2
y
2
- sin x
0

/2
3/2
2
-/2
-
-3/2
-2
y
3
sin x + 2
0

/2
3/2
2
-/2
-
-3/2
-2
y
4
Sin (x + /4)
-/4
/4
3/4
5/4
7/4
-3/4
-5/4
-7/4
-9/4
y
1, 2.
3.
4.
Function
Symbol
Domain
Range
Period
Cosine
y = f(x) = cos x
[-1, 1]
(-, )
2
The cosine function is even; thus it is symmetric to the y-axis and cos (-) = cos . It has xintercepts at odd multiples of /2. The period of trig functions is affected by ω (omega). The
period of cos (ωx) = 2/ω. If ω > 1, the period will be compressed horizontally by a factor of ω. The
amplitude (range) of A cos  is between -A and A, inclusive.
Determine the amplitude and period of each function without graphing:
5. y = -3 cos (½x)
6. y  4 cos(3 x )
7.
9
 3 
3
6/29/2017
y
2
sin  
x
 2 
Use transformations to graph the following:
8
cos x
0

/2
y
9
cos 2x
0
/4
/2
y
10
2cos x
0

/2
y
11
0

cos (x - /2)
/2
y
8, 9.
10.
3/2
2
-/2
-
-3/2
-2
3/4

-/4
-/2
-3/4
-
3/2
2
-/2
-
-3/2
-2
3/2
2
-/2
-
-3/2
-2
11.
Sinusoidal Graphs: The graph of sin x = cos (x - /2). Because of this similarity, the graphs of the
two functions are called sinusoidal. Because the sine function is odd, sin(-ωx) =
-sin(ωx). The cosine function is even; therefore, cos(-ωx) = cos (ωx). To graph a sinusoidal function
like y = A sin (ωx) or y = A cos (ωx), use the amplitude A to determine the minimum and
maximum values of the function. The period is used the divide the x-axis into 4 subintervals.
Then extend the graph in either direction to make it complete.
12. Graph: y = 4 cos (6x)
13. Graph: y = 2 sin (x)
14. Find the equation:
Write the equation of a sine function that has the given characteristics:
15.
Amplitude: 3
16.
Amplitude: 4/3
Period: ½
Period: 3
6/29/2017