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)