Fig1.1 - I-Shou University
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Transcript Fig1.1 - I-Shou University
Chapter 8 Trigonometric Functions
8.1
8.2
8.3
8.4
Radian Measure of Angles
The Sine, Cosine, and Tangent
Derivatives of Trigonometric Functions
Integrals of Trigonometric Functions
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Radian Measure of Angles
There are two methods of measuring angles─by degrees and by radians.
In measuring by degrees, we arbitrarily divide the full circle into 360
equal segments.
1
The angle shown in Figure 8.1.1 cuts off of the circle, and we assign it 45 degrees,
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A more natural method is to use the length of the arc cut off by the angle on a circle of
radius 1. In Figure 8.1.2, s is the length of arc cut off by the angle .
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Radian Measure of Angles
Definition 8.1.1
【數】弧度
The radian measure of an angle is the length of the arc cut off on a circle of
radius 1 by the sides of the angle with its vertex at the center of the circle.
360 degrees 2 radians,
1 degrees
180
radians
and
1 radian
180
degrees.
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Radian Measure of Angles
If the movement is in the counterclockwise direction, we assign a positive sign to
the radian measure. If it is in the clockwise direction, we assign a negative sign.
Figure 8.1.4 shows the cases of 4 and 4 radians.
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Radian Measure of Angles
The angles shown in Figure 8.1.5 illustrate a few simple examples.
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The Sine, Cosine, and Tangent
The sine, cosine, and tangent functions are defined in elementary trigonometry
by means of right triangles. We place an acute angle at the base of a right
銳角
triangle, as shown in Figure 8.2.1.
Then we define the sine, cosine, and tangent of as follows:
The Sine Function
sin
O
.
H
The Cosine Function
co s
A
.
H
The Tangent Function
tan
O
.
A
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The Sine, Cosine, and Tangent
We can use right triangles to find the values of the
trigonometric functions for certain special angles.
(as shown n Figure 8.2.3)
1
1
, tan 1 .
, cos
sin
4
4
4
2
2
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The Sine, Cosine, and Tangent
If we draw a line from the top vertex perpendicular to the base,
as shown in Figure 8.2.4, the line divides the original triangle
into two right triangles, each with acute angles 3 and 6 .
sin
3
3
1
, cos , tan 3 .
2
3 2
3
1
3
1
.
sin , cos
, tan
6 2
6
2
6
3
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The Sine, Cosine, and Tangent
The sine and cosine as coordinates of points on the circle
A more general way to define the trigonometric functions is to use a
circle of radius 1.
If ( x, y ) are the coordinates of the tip of the terminal side, as shown in
Figure 8.2.5, then
sin y, co s x, tan
y
x
.
tan
sin
.
cos
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The Sine, Cosine, and Tangent
Example 8.2.1
Find sin , co s , and tan for 0, , 2, and 3 2.
Solution
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The Sine, Cosine, and Tangent
TABLE 8.2.1
(0, 2)
sin
co s
tan
+
+
+
( 2, )
( ,3 2)
(3 2, 2 )
+
-
+
+
-
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The Sine, Cosine, and Tangent
The trigonometric functions are periodic, with period 2 .In other words,
sin(2 ) sin cos(2 ) cos tan(2 ) tan .
We see that if two angles have radian measures and respectively, their
terminal points are reflections of one another across the horizontal axis.
sin( ) sin cos ( ) cos tan( ) tan .
Another important property of the sin and cosine is the following:
sin 2 cos 2 1 .
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The Sine, Cosine, and Tangent
The graphs of sine, cosine, and tangent
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Derivatives of Trigonometric Functions
The derivatives of sin x and cos x are given by the following formulas:
d
sin x cos x
dx
d
co s x sin x.
dx
Example 8.3.1
Find
d
( x sin x)
dx
Solution
Using the product rule, we have
d
d
( x sin x) x sin x 1 sin x x cos x sin x .
dx
dx
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Derivatives of Trigonometric Functions
Example 8.3.4
dy
Find if
dx
(i) y cos2 x
and (ii) y sin( x2 3x) .
Solution
(i) Using the chain rule with u cos x and y u 2 , we obtain
dy dy du
2u sin x 2 cos x sin x.
dx du dx
(ii) Using the chain rule with u x2 3x and y sin u, we get
dy dy du
(2u 3)co s u (2 x 3) cos( x 2 3 x).
dx du dx
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Derivatives of Trigonometric Functions
The other trigonometric functions
There are three additional trigonometric functions, the secant, cosecant,
and cotangent, defined as follows:
The Secant Function
sec x
1
.
cos x
c s cx
1
.
sin x
The Cosecant Function
The Cotangent Function
1
cos x
cot x
.
tan x sin x
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Derivatives of Trigonometric Functions
d
sec x
Example 8.3.6 Find
dx
Solution Using the chain rule, we obtain
d
d
1
1 d
sin x
sec x (
)
cos x
.
2
2
dx
dx cos x
cos x dx
cos x
We can rewrite the formula in the form
d
d
sec x sec x tan x ,
csc x csc x cot x ,
dx
dx
d
d
tan x sec2 x ,
cot x csc2 x .
dx
dx
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Integrals of Trigonometric Functions
From the derivative formulas (15) and (16) we get the
following antiderivative formulas:
sin xdx cos x c
cos xdx sin x c
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Integrals of Trigonometric Functions
Example 8.4.1
Find the area under the cosine curve between 2 and 2 (shown in Figure 8.4.1).
Solution
Using the antiderivative formula (41) and the fundamental theorem of calculus, we
get
2
2
cos xdx sin x
2
2
sin
sin (- ) 2
2
2
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Integrals of Trigonometric Functions
Example 8.4.3 Find (i) sin 3 t cos tdt , (ii)
2
t
cos(
t
1)dt ,
(iii)
2
0
t cos tdt .
Solution
(i)Using the substitution u sin t so that du dt cos t , we get
u4
sin 4 t
sin t cos tdt u du 4 c 4 c
3
3
(ii)Using the substitution u t 2 1, we have du 2tdt . Therefore
1
1
1
2
2
t
cos(
t
1))
dt
cos
udu
sin
u
c
sin(
t
1) c
2
2
2
(iii)We use integration by parts, with u t and dv dt cos t . Then du dt 1 and
v sin t , so that
2
0
t cos tdt t sin t
2
0
2
0
sin tdt
2
cos t
2
0
2
1.
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Integrals of Trigonometric Functions
Example 8. 4. 4 Find
tan xdx.
sin x
dx,
Solution Writing tan xdx
cos x
We make the substitution u cos x, so that du dx sin x. Therefore
1
xdx
tan
u du ln u c ln cos x c.
So that
tan xdx ln sec x c.
Similarly
cot xdx ln csc x c.
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Integrals of Trigonometric Functions
Example 8. 4. 5 Find
sec xdx
Solution We first change the form of the integrand by multiplying and dividing by the
same expression:
sec x tan x
sec2 x sec x tan x
sec xdx sec x sec x tan x dx sec x tan x dx.
Letting u tan x sec x and du (sec2 x sec x tan x)dx, we obtain
sec xdx
Therefore,
du
ln u c.
u
+
sec xdx ln tan x sec x c.
Similarly
csc xdx ln cot x csc x c.
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