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Sketching Combinations of Trigonometric Functions
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Graphs of Trigonometric Functions
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2002 Doug MacLean
Note: For most problems, the graph can be viewed interactively using Java applets with Netscape Communicator or
Internet Explorer.
Colour Conventions: The graph of the function being studied is drawn in black, that of the first derivative is in red ,
and that of the second derivative is in blue . The asymptotes are drawn in green , except when they coincide with the
axes, which are black.
The identity a sin x + b cos x = A sin(x − φ) where A =
√
b
a2 + b2 and φ = − arctan
a
is easily verified using the angle sum identity for the sine function.
Graphs of Trigonometric Functions
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Example 1:
4
0.93
f (x) = 3 sin x − 4 cos x = 5 sin(x − φ) where φ = arctan
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π
2
+ nπ
and f (x) = −3 sin x + 4 cos x = −5 sin(x − φ) = 0 if x = φ + nπ .
The “interesting” values are thus φ + nπ , φ +
Step 2:
π
2
+ nπ .
Put these values of x into increasing order.
Step 3:
Put together as good a table as you can showing the signs of f (x) and f (x) on the intervals into which
the interesting values divide the domain of f .
x
0
f (x) 4
f (x)
3
f (x)
−4
Step 4:
half-frowns.
(0, φ) φ
+
+
−
0
5
0
π
φ, φ +
2
−
+
+
π
2
−5
0
5
φ+
φ+
π
,φ
2
−
−
+
+π
φ+π
0
−5
0
φ + π, φ +
+
−
−
3π
2
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Step 1:
f (x) = 3 cos x + 4 sin x = 5 cos(x − φ) = 0 if x = φ +
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φ+
3π
2
5
0
−5
φ+
3π
, 2π
2
+
+
−
2π
4
3
−4
Plot the “interesting points” and connect them with curves which are either left or right half-smiles or
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5
3
2
1
-1
-2
-3
-4
-5
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0
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Graphs of Trigonometric Functions
φ
π/2
π
3π/2
2π
x
Graphs of Trigonometric Functions
Example 2:
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f (x) = 2 sin x + sin x = sin x(2 + sin x).
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e w ane
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Since f has period 2π , we need only sketch the graph over the interval [0, 2π ].
Step 1:
f (x) = 2 cos x + 2 sin x cos x = 2 cos x(1 + sin x) = 0 if cos x = 0, i.e., x =
π
2
+ nπ , or sin x = −1, i.e., x =
3π
2
+ 2nπ
and f (x) = −2 sin x + 2(1 − 2 sin2 x) = −4 sin2 x − 2 sin x + 2 = −2(2 sin2 x + sin x − 1) = (2 sin x − 1)(sin x + 1) = 0
3π
1
π
5π
if sin x = −1, i.e., x =
+ 2nπ , or sin x = , i.e. x =
+ nπ ,
+ nπ ..
2
2
6
6
π π 5π 3π
The “interesting” values in [0, 2π ] are thus , ,
,
.
6 2 6
2
Step 2:
Put these values of x into increasing order:
π π 5π 3π
, ,
,
.
6 2 6
2
Step 3:
Put together as good a table as you can showing the signs of f (x) and f (x) on the intervals into which
the interesting values divide the domain of f .
x
0
f (x) 2
f (x) 2
f (x)
0
π
6
+
+
+
0,
π
6
0
+
3
π π
,
6 2
−
+
+
π
2
−4
0
+
π 5π
,
2 6
−
−
+
5π
6
0
−
+
5π 3π
,
6
2
+
−
3π
2
0
0
−1
3π
, 2π
2
+
+
−
2π
2
2
0
half-frowns.
Plot the “interesting points” and connect them with curves which are either left or right half-smiles or
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Step 4:
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Graphs of Trigonometric Functions
e w ane
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2002 Doug MacLean
y
4
3
2
1
0
-1
-2
-3
-4
π/2
π
3π/2
2π
x
Graphs of Trigonometric Functions
f (x) = sin 2x − 2 sin x = 2 sin x cos x − 2 sin x = 2 sin x(1 − cos x)
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Example 3:
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e w ane
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Since f has period 2π , we need only sketch the graph over the interval [0, 2π ].
Step 1:
f (x) = 2 cos 2x − 2 cos x = 2(2 cos2 x − 1) − 2 cos x = 2(2 cos2 x − cos x − 1) = 2(2 cos x + 1)(cos x − 1) = 0 if cos x = 1,
1
2π
4π
i.e., x = 2nπ , or cos x = − 2 , i.e., x = 3 + 2nπ , 3 + 2nπ
and f (x) = −4 sin 2x + 2 sin x = −8 sin x cos x + 2 sin x = −2 sin x(4 cos x − 1) = 0 if sin x = 0, i.e., x = nπ , or
1
cos x = , i.e. x = φ + 2nπ , −φ + (2n + 1)π , where φ = arccos 14 1.32.
4
2π 4π
The “interesting” values in [0, 2π ] are thus 0, π ,
,
, φ, 2π − φ.
3
3
Step 2:
Put these values of x into increasing order:
0, φ,
4π
2π
, π,
, 2π − φ .
3
3
Step 3:
Put together as good a table as you can showing the signs of f (x) and f (x) on the intervals into which
the interesting values divide the domain of f .
x
0 (0, φ) φ
f (x) 0
f (x) 0
f (x)
0
−
−
−
0
−
−
2π
φ,
3
+
−
−
2π
3
+
0
−
2π
,π
3
+
+
−
π
0
+
0
4π
π,
3
+
+
+
4π
3
+
0
+
4π
, 2π − φ
3
+
−
+
2π − φ (2π − φ, 2π ) 2π
0
−
+
+
−
+
0
0
0
Plot the “interesting points” and connect them with curves which are either left or right half-smiles or
half-frowns.
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Step 4:
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Graphs of Trigonometric Functions
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2002 Doug MacLean
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5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
φ
π/2
π
3π/2
2π
x
Graphs of Trigonometric Functions
Example 4:
2
2
2
2
Since f has period 2π , we need only sketch the graph over the interval [0, 2π ].
Step 1:
f (x) = 2(cos x − 1) sin x = 2 sin x cos x − 2 sin x = 0 if cos x = 1, i.e., x = 2nπ , or sin x = 0, i.e., x = nπ 0
and f (x) = 2 cos 2x − 2 cos x = 2(2 cos2 x − cos x − 1) = 2(2 cos x + 1)(cos x − 1) = 0 if cos x = 1, i.e., x = 2nπ , or
1
2π
4π
cos x = − , i.e. x =
+ 2nπ ,
+ 2nπ .
2
3
3
2π 4π
The “interesting” values in [0, 2π ] are thus 0, π ,
,
, 2π .
3
3
Put these values of x into increasing order:
0,
4π
2π
, π,
, 2π .
3
3
Step 3:
Put together as good a table as you can showing the signs of f (x) and f (x) on the intervals into which
the interesting values divide the domain of f .
x
0
f (x) 0
f (x) 0
f (x)
2
2π
0,
3
−
−
+
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2002 Doug MacLean
2 − (cos x − 1)
Step 2:
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f (x) = 2 cos x + sin x = 2 cos x + 1 − cos x = −(cos x − 2 cos x − 1) = − (cos x − 1) − 2 =
2
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2π
3
0
−
2π
,π
3
+
−
π
+
0
−2
4π
π,
3
+
+
4π
3
0
+
4π
, 2π
3
−
+
+
2π
0
0
2
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Plot the “interesting points” and connect them with curves which are either left or right half-smiles or
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Step 4:
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half-frowns.
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Graphs of Trigonometric Functions
e w ane
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2002 Doug MacLean
y
4
3
2
1
0
-1
-2
-3
-4
π/2
π
3π/2
2π
x
Graphs of Trigonometric Functions
f (x) = sin 3x + cos 4x
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Example 5:
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e w ane
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2002 Doug MacLean
Step 1:
f (x) = 3 cos 3x − 4 sin 4x and f (x) = −9 sin 3x − 16 cos 4x .
Some of the roots of these functions may be partially solved using trigonometric identities, but not easily:
3 cos(2x + x) − 4 sin 2(2x) = [32 sin3 x − 12 sin2 x − 16 sin x + 3] cos x = 0 if cos x = 0, i.e., x =
π 3π
,
2 2
or if 32 sin3 x − 12 sin2 x − 16 sin x + 31 = 0, which requires us to solve the cubic equation 32z3 − 12z2 − 16z + 3 = 0
which has three roots between −1 and 1, none of which is a rational number.
It turns out that both f (x) and f (x) have 8 roots in the interval [0, 2π ], but this is most easily discovered by computer
1
generation of their graphs: ( Judicious scaling helps: in the following diagram, we show the graph of 16 f (x) in blue
and that of 14 f (x) in red .)
y
2
1
0
-1
-2
π/2
π
3π/2
2π
x
f (x) = x sin x
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Example 6:
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Graphs of Trigonometric Functions
e w ane
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2002 Doug MacLean
f (x) = sin x + x cos x
y
7
and f (x) = sin x + cos x − x sin x .
6
The “interesting” values are not easily computed.
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
π/2
π
3π/2
2π
x