1 chapter 8 trigonometric functions 8.1 radian measure of angles 8.2 the sine, cosine, and tangent...

1

Chapter 8 Trigonometric Functions

8.1 Radian Measure of Angles

8.2 The Sine, Cosine, and Tangent

8.3 Derivatives of Trigonometric Functions

8.4 Integrals of Trigonometric Functions

2

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.

The angle shown in Figure 8.1.1 cuts off1

8of the circle, and we assign it 45 degrees,

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 .

3


Definition 8.1.1 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 degrees180

radians and 1 radian

180

degrees.

【數】弧度

4


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.

5


The angles shown in Figure 8.1.5 illustrate a few simple examples.

6

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

triangle, as shown in Figure 8.2.1.

Then we define the sine, cosine, and tangent of as follows:

The Sine Function

sinO

H .

The Cosine Function

sA

coH

.

The Tangent Function

tanO

A .

銳角

7


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 1sin , cos , tan 1

4 4 42 2

.

8


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 .

3 1sin , cos , tan 3

3 2 3 2 3

.

1 3 1sin , cos , tan

6 2 6 2 6 3

.

9


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 , s , tany

y co xx

.

sintan

cos

.

10


Example 8.2.1 Find sin , sco , and tan for 0, , 2, and 3 2.

Solution

11


TABLE 8.2.1

(0, 2) ( 2, ) ( ,3 2) (3 2,2 )

sin

sco

tan

+

+

+

+

-

-

-

-

+

-

+

-

12


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:

2 2sin cos 1 .

13


The graphs of sine, cosine, and tangent

14

Derivatives of Trigonometric Functions

The derivatives of sin x and cos x are given by the following formulas:

sin cosd

x xdx

s sin .dco x x

dx

Example 8.3.1

Find ( sin )dx x

dx

Solution

Using the product rule, we have

( sin ) sin 1 sin cos sind dx x x x x x x x

dx dx .

15


Example 8.3.4

Finddy

dxif

(i) 2cosy x and (ii) 2sin( 3 )y x x .

Solution

(i) Using the chain rule with cosu x and 2y u , we obtain

2 sin 2cos sin .dy dy du

u x x xdx du dx

(ii) Using the chain rule with 2 3u x x and sin ,y u we get

2(2 3) s (2 3)cos( 3 ).dy dy du

u co u x x xdx du dx

16


The other trigonometric functions

There are three additional trigonometric functions, the secant, cosecant,

and cotangent, defined as follows:

The Secant Function

1sec

cosx

x .

The Cosecant Function

1s

sinc cx

x .

The Cotangent Function

1 coscot

tan sin

xx

x x .

17


Example 8.3.6 Find secd

xdx

Solution Using the chain rule, we obtain

2 2

1 1 sinsec ( ) cos

cos cos cos

d d d xx x

dx dx x x dx x .

We can rewrite the formula in the form

sec sec tand

x x xdx

, csc csc cotd

x x xdx

,

2tan secd

x xdx

, 2cot cscd

x xdx

.

18

Integrals of Trigonometric Functions

From the derivative formulas (15) and (16) we get the

following antiderivative formulas:

sin cosxdx x c

cos sinxdx x c

19


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

2 222

cos sin sin sin (- ) 22 2

xdx x

20


Example 8.4.3 Find (i) 3sin cost tdt , (ii) 2cos( 1)t t dt , (iii)2

0cost tdt

.

Solution

(i)Using the substitution sinu t so that cosdu dt t , we get 4 4

3 3 sinsin cos

4 4

u tt tdt u du c c

(ii)Using the substitution 2 1u t , we have 2du tdt . Therefore

2 21 1 1cos( 1)) cos sin sin( 1)

2 2 2t t dt udu u c t c

(iii)We use integration by parts, withu t and cosdv dt t . Then 1du dt and

sinv t , so that

2 2200 0

cos sin sint tdt t t tdt 2

0cos 12 2

t .

21


Example 8. 4. 4 Find tan .xdx

Solution Writing sin

tan ,cos

xxdx dx

x

We make the substitution cos ,u x so that sin .du dx x Therefore

1tan ln ln cos .xdx du u c x c

u

So that tan ln sec .xdx x c

Similarly

cot ln csc .xdx x c

22


Example 8. 4. 5 Find sec xdx

Solution We first change the form of the integrand by multiplying and dividing by the

same expression: 2sec tan sec sec tan

sec sec .sec tan sec tan

x x x x xxdx x dx dx

x x x x

Letting tan secu x x and 2(sec sec tan ) ,du x x x dx we obtain

sec ln .du

xdx u cu

Therefore,

sec ln tan sec .xdx x x c

Similarly

csc ln cot csc .xdx x x c

+

1 chapter 8 trigonometric functions 8.1 radian measure of angles 8.2 the sine, cosine, and tangent...

Documents