Graphs of the Sine, Cosine, & Tangent Functions
Objectives:
1. Graph the sine, cosine, & tangent functions.
2. State all the values in the domain of a basic trigonometric function that correspond to a given value of the range.
3. Graph the transformations of sine, cosine, & tangent functions.
7.1
Graphing the Cosine Function on the Coordinate PlaneDegrees Radians cos(t
)0° 0 1
30° .86645° .70760° .590° 0120° -.5135° -.707150° -.866180° -1210° -.866225° -.707240° -.5270° 0300° .5315° .707330° .866360° 1
6
4
3
2
32
43
65
67
34
45
23
35
47
611
2
Graphing the Sine Functionon the Coordinate Plane
Characteristics of the Sine & Cosine Functions
Period : 2πDomain: The set of all real numbers (−∞, ∞)Range: [−1, 1] Function Type: Sine (Odd)
Cosine (Even)
The period of a function is the amount of time or length of a complete cycle. In other words, how long until the graph starts repeating. For the sine and cosine functions, the period is the same.
Remember: Even Functions are symmetric about the y-axis, Odd Functions are symmetric about the origin (shown below).
Example #1 State all values of t for which sin(t) = 1.
Remember that sine, the y-coordinate, is 1 at 90°. Any angle coterminal with that is also a solution.
(1,0)
(0,1)
(0,-1)
(-1,0) 0°, 360°
90°
180°
270°
Example #2 State all the values of t for which cos(t) =
21
Remember that cosine, the x-coordinate, is -½ at 120° and 240°. Any angle coterminal with those are also a solutions.
(1,0)
(0,1)
(0,-1)
(-1,0) 0°, 360°
90°
180°
270°
Graphing the Tangent Functionon the Coordinate Plane
Characteristics of the Tangent Function
Period: πDomain: All real numbers except odd multiples ofRange: All real numbers (−∞, ∞)Function Type: Odd
2
Example #3 State all values of t for which tan(t) = 1.
Tangent is 1 where sine and cosine values are the same. This occurs at 45° and 225°.
The cycle is shorter for tangent though, so to specify all solutions we only need to add 180° to our original solution.
Basic Transformations of Sine, Cosine, & Tangent Vertical Stretches
Vertical stretches or compressions by a factor of “a”.
ReflectionsReflections over the x-axis.
Vertical ShiftsVertical shifting of “b” units.
tatftatftatf tan)(cos)(sin)(
ttfttfttf tan)(cos)(sin)(
bttfbttfbttf tan)(cos)(sin)(
Example #4 List the transformation(s) and sketch the
graph. ttg cos2)(
Vertical stretch by a factor of 2.
t
1
2
3
–1
–2
–3
g( t)
Example #5 List the transformation(s) and sketch the
graph.ttg sin
31)(
Vertical compression by a factor of 1/3 and x-axis reflection. t
0.5
1
1.5
–0.5
–1
–1.5
g( t)
Example #6 List the transformation(s) and sketch the
graph. 4tan)( ttg
Vertical shift of four units down.
t
2
4
6
8
10
–2
–4
–6
–8
–10
g( t)