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Trigonometric Functions

Trigonometric Functions Review

Angles: Degree-Radian Measure

In a plane, an angle is formed by rotating a ray, called the initial side of the angle, around its endpoint until the ray coincides with another ray, called the terminal side of the angle. The common endpoint of these rays is called the vertex.

A counterclockwise rotation produces a positive angle, and a clockwise rotation produces a negative angle. Two different angles with the same initial and terminal sides are said to be coterminal.

There are two widely used measures of angles: the degree and the radian.

Degree-Radian Conversion: deg

180rad

rad

.

Example (From Degrees to Radians). Find the radian measure of 1 .

Solution. 1

, 0.0175180 180

radrad rad rad

rad

.

A comparison of degree and radian measure for a few important angles is given in the following table:

Radian 0 / 6 / 4 / 3 / 2 2

Degree 0 30 45 60 90 180 360

An angle in a rectangular coordinate system is said to be in standard position if its vertex is at the origin and its initial side is on the positive x axis.

Trigonometric Functions

Consider a unit circle (radius 1) in a coordinate system with center at the origin. The terminal side of any angle in standard position will pass through this circle at some point. The abscissa of that point is called the cosine of (abbreviated cos ), and the ordinate of the point is the sine of (abbreviated sin ). The set of all ordered pairs of the form ( , cos ) and the set of all ordered pairs of the form ( , sin ) constitute, respectively, the cosine and sine functions. The domain of these two functions is the set of all angles, positive or negative, with measure either in degrees or radians. The range is a subset of the set of real numbers.

It is necessary for our work in calculus to define these two trigonometric functions in terms of real-number domains. This is done as follows:

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Definition (Sine and Cosine Functions with Real-Number Domains). For any real number x,

sin sin( )x x radians and cos cos( )x x radians .

Since cos x and sin x are the coordinates of a point on the unit circle,1 sin 1x and 1 cos 1x .

As x Increases from siny x cosy x

0 to / 2 Increases fro 0 to 1 Decreases from 1 to 0

/ 2 to Decreases from 1 to 0 Decreases from 0 to -1

to 3 / 2 Decreases from 0 to -1 Increases from -1 to 0

3 / 2 to 2 Increases from -1 to 0 Increases fro 0 to 1

It can be shown that sin( 2 ) sin , cos( 2 ) cosx x x x , for all real numbers x.

Functions such that ( ) ( )f x p f x for some positive constant p and all real numbers xfor which the functions are defined are said to be periodic. The smallest such value of p is called the period of the function.

Both the sine and cosine functions are periodic with period 2 .

It can be shown that the sine and cosine functions are continuous for all real numbers.

Four Other Trigonometric Functions

The other four trigonometric functions are the tangent, cotangent, secant, and cosecant.

Definition (Four Other Trigonometric Functions).

sin 1tan , cos 0; sec , cos 0;

cos coscos 1

cot , sin 0; csc , sin 0.sin sin

xx x x x

x xx

x x x xx x

The functions tan x and cot x have period . The other four trigonometric functions -sin x , cos x , sec x , and csc x - all have period 2 .

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Derivatives of Trigonometric Functions

Derivative Formulas

From the definition of the derivative,

0

sin( ) sinsin lim

h

d x h xx

dx h

.

On the basis of trigonometric identities and some special trigonometric limits, it can be shown that the limit on the right is cos x , i.e.,

sin cosd

x xdx

.

Similarly, it can be shown that

cos sind

x xdx

.

Derivatives of Sine and Cosine.

Basic Form:

sin cos , cos sind d

x x x xdx dx

.

Generalized Form:

For ( )u u x ,

sin cos , cos sind du d du

u u u udx dx dx dx

.

Examples.

(A) 2 2 2 2sin cos 2 cosd d

x x x x xdx dx

;

(B) cos(2 5) sin(2 5) (2 5) 2sin(2 5)d d

x x x xdx dx

;

(C) 2 2 2 2(3 )cos (3 ) cos cos (3 ) ( 3 )sin (6 1) cosd d d

x x x x x x x x x x x x x xdx dx dx

;

(D) 1 22

1 sinsec (cos ) (cos ) cos tan sec

cos (cos )

d d d d xx x x x x x

dx dx x dx dx x .

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Application

Example (Revenue). Revenues from the sale of ski jackets are given approximately by

( ) 1.55 1.45cos , 0 10426

tR t t

,

where R(t) is revenue (in thousands of dollars) for a week of sales t weeks after January 1.

(A) What is the rate of change of revenue t weeks after the first of the year?

(B) What is the rate of change of revenue 10 weeks after the first of the year? 26 weeks after the first of the year? 40 weeks after the first of the year?

(C) Find all local maxima and minima for 0 104t .

(D) Find the absolute maximum and minimum for 0 104t .

Solution.

(A) 1.45

( ) sin , 0 10426 26

tR t t

.

(B) (10) $0.164R thousand, or -$ 164 per week;

(26) $0R per week;

(40) $0.174R thousand, or $ 174 per week.

(C) Find the critical points:

1.45( ) sin 0, 0 104,

26 26

sin 0,26

, 2 , 3 ( : 0 104 0 4 ),26 26

26, 52, 78.

tR t t

t

t tNote t implies that

t

Differentiate ( )R t to get ( )R t :2

2

1.45( ) cos

26 26

tR t

.

We use the second-derivative test to get the results shown in the following table:

t ( )R t Graph of R

26 Local minimum

52 Local maximum

78 Local minimum

(D) Evaluate ( )R t at endpoints t = 0 and t = 104 and at the critical points found in part (C):(0) $3000R Absolute maximum;

(26) $100R Absolute minimum;(52) $3000R Absolute maximum;(78) $100R Absolute minimum;(104) $3000R Absolute maximum.

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Integration of Trigonometric Functions

Integral Formulas

cos sin , sin cosx dx x C x dx x C .

Example (Area Under a Sine Curve). Find the area under the sine curve siny x from 0 to .

Solution.

Area0

0

sin cos (cos cos0) ( 1 1) 2x dx x

.

General Integral Formulas:

For ( )u u x ,

cos sin , sin cosu du u C u du u C .

Examples.

(A) 2

2 2 21 1 1 1sin 2 sin sin cos cos

2 2 2 22

u xx x dx x x dx u du u C x C

du xdx

;

(B) 6 6

5 5sin (sin )(sin ) cos

cos 6 6

u x u xx x dx u du C C

du x dx

.

Application

Example (Total Revenue). Revenues from the sale of ski jackets are given approximately by

( ) 1.55 1.45cos , 0 10426

tR t t

,

where R(t) is revenue (in thousands of dollars) for a week of sales t weeks after January 1. Find the total revenue taken in over the 2-year period – that is, from t = 0 to t = 104.

Solution. The area under the graph of the revenue equation for the 2-year period approximates the total revenue taken in for that period. This area (and therefore the total revenue) is given by the following definite integral:

Total revenue104104

0 0

261.55 1.45cos 1.55 1.45 sin $161.2

26 26

t tdt t

thousand.