Digital Lesson Graphs of Trigonometric Functions Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos.
Download ReportTranscript Digital Lesson Graphs of Trigonometric Functions Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos.
Digital Lesson
Graphs of Trigonometric Functions
Properties of Sine and Cosine Functions The graphs of
y
= sin
x
and
y
= cos
x
have similar properties: 1. The domain is the set of real numbers.
2. The range is the set of
y
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
5. Each function cycles through all the values of the range over an
x
6. The cycle repeats itself indefinitely in both directions of the
x
-axis.
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2
Graph of the Sine Function To sketch the graph of
y
= sin
x
first locate the key points.
These are the maximum points, the minimum points, and the intercepts.
x
0 3 2 sin
x
0 2 1 0 2 -1 0 Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a
period
.
3 2 2 1
y
2
y
= sin
x
3 2 2 5 2
x
1
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3
Graph of the Cosine Function To sketch the graph of
y
= cos
x
first locate the key points.
These are the maximum points, the minimum points, and the intercepts.
x
cos
x
0 1 2 0 -1 3 2 0 2 1 Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a
period
.
3 2 2 1
y
2
y
= cos
x
3 2 2 5 2
x
1
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4
Example
: Sketch the graph of
y
= 3 cos
x
on the interval [– , 4 ].
Partition the interval [0, 2 ] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.
y x
= 3 cos
x
0 3 2 0 -3 3 0 2 2 3 max
x
-int min max (0, 3)
y
2 1 2
x
-int 3 4
x
1 2 3 ( , 0) 2 2
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5
The
amplitude
of
y
=
a
sin
x
(or
y
=
a
cos
x
) is half the distance between the maximum and minimum values of the function.
amplitude = |
a
| If |
a
| > 1, the amplitude stretches the graph vertically.
If 0 < |
a
| > 1, the amplitude shrinks the graph vertically.
If
a
< 0, the graph is reflected in the
x
-axis.
y
4
y
= sin
x
2 3 2 2
x y y
1 = sin
x
2 = – 4 sin
x
reflection of
y
= 4 sin
x
4
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y
= 2 sin
x y
= 4 sin
x
6
The
period
of a function is the
x
interval needed for the function to complete one cycle.
For
b
0, the
period
of
y
=
a
sin
bx
is .
b
For
b
If 0 <
y
b
0, the
period
of
y
=
a
cos
bx
is also .
b
< 1, the graph of the function is stretched horizontally.
y
sin period: 2
y
period: 2 sin
x
x
2 If
b
> 1, the graph of the function is shrunk horizontally.
y
cos 1 period: 4
x
y
2 3 4
y x
cos
x
period: 2
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7
Use basic trigonometric identities to graph
y
=
f
(–
x
)
Example 1
: Sketch the graph of
y
= sin (–
x
).
The graph of
y
= sin (–
x
) is the graph of
y
= sin
x
reflected in the
x
-axis.
y y
= sin (–
x
) Use the identity sin (–
x
) = – sin
x y
= sin
x
2
x
Example 2
: Sketch the graph of
y
= cos (–
x
).
The graph of
y
= cos (–
x
) is identical to the graph of
y y
= cos
x.
Use the identity cos (–
x
) = – cos
x x
= cos ( –
x
) ) 2
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8
Example
: Sketch the graph of
y
= 2 sin (–3
x
).
Rewrite the function in the form
y
=
a
sin
bx
with
b
> 0 Use the identity sin (–
x
) = – sin
x:
amplitude : |
a
| = | –2 | = 2 Calculate the five key points.
x y
= –2 sin 3
x
6 2
y
6
y
= 0 6 3 0 3 –2 0 ( , 2) 2 2 2 3 2 period : sin (– 3
x
) = – 2 5 6 2 2 2
b
= 2 3 0 3
x
sin 3
x
(0, 0) 2 ( ,-2) 6 ( , 0) 3 ( , 0) 3
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9
Graph of the Tangent Function To graph
y
= tan
x
sin
x
, use the identity .
x
At values of
x
for which cos
x
tan cos = 0, the tangent function is undefined and its graph has vertical asymptotes.
y
Properties of
y
= tan
x
1. domain : all real
x x
k
k
2 2. range: (– , + ) 3. period: 3 2 2 2
x
k
2
k
period:
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3 2
x
10
Example
: Find the period and asymptotes and sketch the graph of
y
1 tan 2
x x
y x
3 4 4 1. Period of Period
y
of = tan
y
x
is . tan 2
x
is 2 .
2. Find consecutive vertical 2
x
2 , 2
x
Vertical asymptotes: 2
x
x
4 : ,
x
4 3 8 8 , 1 3 8 , 1 3 3 8 , 2 1 3 3. Plot several points in ( 0 , 2 ) 4. Sketch one branch and repeat.
y
x
1 3 tan 2
x
8 1 3 0 0 8 1 3 3 8 1 3
x
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11
Graph of the Cotangent Function To graph
y
= cot At values of
x x
for which sin
x
cot
x
cos sin
x x
= 0, the cotangent function is undefined and its graph has vertical asymptotes.
y
Properties of
y
= cot
x y
cot
x
1. domain : all real
x x
k
k
2. range: (– , + ) 3. period: 4. vertical asymptotes:
x
k
k
3 2 2 2 3 2 2
x
vertical asymptotes
x
x
0
x
x
2
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12
Graph of the Secant Function The graph
y
= sec
x
sec
x
1 cos
x
At values of
x
for which cos
x
= 0, the secant function is undefined and its graph has vertical asymptotes.
y y
sec
x
Properties of
y
= sec
x
1. domain : all real
x x
k
(
k
) 2 2. range: (– ,–1] 3. period: [1, + ) 4. vertical asymptotes:
x
k
2
k
2 4 4 2 3 2 2 5 2
y
cos
x x
3
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13
Graph of the Cosecant Function To graph
y
= csc
x
csc
x
1 sin
x
At values of
x
for which sin
x
= 0, the cosecant function is undefined and its graph has vertical asymptotes.
y
Properties of
y
= csc
x
4 1. domain : all real
x x
k
k
2. range: (– ,–1] [1, + ) 3. period: 4. vertical asymptotes:
x
k
k
2 2 where sine is zero.
4 3 2 2
y
csc
x x
5 2
y
sin
x
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14