SELF-CHECK DIAGNOSTIC TEST - Tan calculus early tr- 123docz.net

1. Given that and , find .

2. Determine whether the function is even, odd, or neither.

3. Verify the identity .

4. Using the substitution , where and , express in terms of .

5. Solve the equation , where .

Answers to Self-Check Diagnostic Test 0.3 can be found on page ANS 2.

0 u2p

cosu2 sin2u10 a2x2 u

p2 u p2 a0

xasinu cotx1

1tanxcotx

f(x) sin 2x

21cos2x1 tanu

0 up2

secu53

FIGURE 1

x y

(b) A positive angle in standard position (c) A negative angle in standard position (a) An angle

Initial side

Initial side Vertex

Terminal side

Terminal side Terminal side

¨ ¨

In a rectangular coordinate system an angle (the Greek theta) is in standard posi- tionif its vertex is centered at the origin and its initial side coincides with the positive -axis. An angle is positiveif it is generated by a counterclockwise rotation and neg- ativeif it is generated by a clockwise rotation (Figure 1b–c).

Radian Measure of Angles

We can express the magnitude of an angle in either degrees or radians. In calculus, however, we prefer to use the radian measure of an angle because it simplifies our work.

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DEFINITION Radian Measure of an Angle

If is the length of the arc subtended by a central angle in a circle of radius , then

(1) is the radian measureof (see Figure 2).u

u s r r

s u

For convenience we often work with the unit circle, that is, the circle of radius 1 centered at the origin. On the unit circle, an angle of 1 radian is subtended by an arc of length 1 (see Figure 3). To specify the units of measure for the angle in Figure 3, we write radian or . By convention, if the unit of measure is not specifi- cally stated, we assume that it is radians.

Since the circumference of the unit circle is and the central angle subtended by one complete revolution is 360°, we see that

(2) and

(3) These relationships suggest the following useful conversion rules.

1° p

180 rad 1 rad a180

p b° 2p radians (rad)360°

2p u1

u1 FIGURE 2 u

FIGURE 3

The unit circle x2y21

Converting Degrees and Radians

To convert degrees to radians, multiply by . To convert radians to degrees, multiply by 180. p p 180

EXAMPLE 1 Convert each of the following to radian measure:

a. 60° b. 300° c.

Solution

a. , or rad

b. , or rad

c. , or rad5p

225ⴢ p 4 180

5p 4

5p 300ⴢ p 3

1805p 3

p 60ⴢ p 3

180 p 3

225°

x y

sArc length

¨ 0

x y

1Arc length

1 1

¨ 0

EXAMPLE 2 Convert each of the following to degree measure:

a. rad b. rad c. rad

Solution

a. , or 60°

b. , or 135°

c. , or

More than one angle may have the same initial and terminal sides. We call such angles coterminal.For example the angle has the same initial and terminal sides as the angle u 2p>3(see Figure 4).

4p>3 315°

7p 4 ⴢ180

p 315

3p 4 ⴢ180

p 135 p

3 ⴢ180 p 60

7p 4 3p

4 p

FIGURE 4 Coterminal angles

An angle may be greater than rad. For example, an angle of rad is generated by rotating a ray in a counterclockwise direction through one and a half revolutions (Figure 5a). Similarly, an angle of radians is generated by rotating a ray in a clockwise direction through one and a quarter revolutions (Figure 5b).

5p>2

3p 2p

x y

(a) ¨

x y

(b) ¨ 4π

4π 3

2π 3

¨ 2π

¨ 3

x y

(a) ¨ 3π

x y

(b) ¨ 5π 2

¨ 3π 5π

¨ 2

FIGURE 5 Angles generated by more

than one revolution

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By rewriting Equation (1), , we obtain the following formula, which gives the length of a circular arc.

us>r

Degrees 0° 30° 45° 60° 90° 120° 135° 150° 180° 270° 360°

Radians 0 p

6 p 4

p 3

p 2

2p 3

3p 4

6 p 3p

2 2p

TABLE 1

Length of a Circular Arc

(4) sru

Another related formula that we will use later in calculus gives the area of a circular sector.

Area of a Circular Sector

(5) A1

2 r2u

EXAMPLE 3 What is the length of the arc subtended by radians in a circle of radius 3? What is the area of the circular sector determined by ?

Solution To find the length of the arc, we use Equation (4) to obtain

The area of the sector is obtained by using Equation (5). Thus,

The Trigonometric Functions

Two approaches are generally used to define the six trigonometric functions. We summarize each approach here.

21p 4 A1

2 r2u 1

2 (3)2a7p 6 b s3a7p

6 b 7p 2

u u7p>6 Note In Equations (4) and (5) must be expressed in radians.u

The radian and degree measures of several common angles are given in Table 1.

Be sure that you familiarize yourself with these values.

THE TRIGONOMETRIC FUNCTIONS The Right Triangle Definition For an acute angle (see Figure 6),

The Unit Circle Definition

Let denote an angle in standard position, and let denote the point where the terminal side of meets the unit circle. (See Figure 7.) Then

, ,

, cotu x, y0

x0 y tanuy

x0 secu 1

y0 x cscu1

cosux sinuy

P(x, y) u

cotu adj tanu opp opp

secu hyp adj adj

cscu hyp cosu adj opp

sinu opp hyp hyp u

FIGURE 6

FIGURE 7 The unit circle

Referring to the point on the unit circle (Figure 7), we see that the coordinates of can also be written in the form

and (6)

Note and are not defined when . Also, and are not defined when .

Table 2 lists the values of the trigonometric functions of certain angles. Since these values occur very frequently in problems involving trigonometry, you will find it helpful to memorize them. The right triangles shown in Figure 8 can be used to help jog your memory.

cotu cscu

x0 secu

tanu

ysinu xcosu

P(x, y)

TABLE 2 (radians)

U U(degrees) sinU cosU tanU

6 30° 1

13 2

13 3 p

4 45° 12

2 1

3 60° 13

2 13

Adjacent side

Hypotenuse Opposite

side

x x y

y P(x, y) 1

30 45

1 1

√3

√2

FIGURE 8

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The sign of a trigonometric function of an angle is determined by the quadrant in which the terminal side of lies. Figure 9 shows a helpful way of remembering the functions that are positive in each quadrant. The signs of the other functions are easy to remember, since they are all negative.

FIGURE 9 The trigonometric functions that are positive in each quadrant can be remembered with the mnemonic

device ASTC:All Students Take Calculus. The functions that are not listed in each quadrant are negative.

FIGURE 10

The next example illustrates how we find the trigonometric functions of an angle.

EXAMPLE 4 Find the sine, cosine, and tangent of .

Solution We first determine the reference angle for the given angle. As is indicated in Figure 11, the reference angle is , or 45°. Since

and the sine is negative in Quadrant III, we conclude that . Sim- ilarly, since and the cosine is negative in Quadrant III, we conclude

that . Finally, since and the tangent is positive in

Quadrant III, we conclude that .

The values of the trigonometric functions that we found in Example 4 are exact.

The approximate value of any trigonometric function can be found by using a calculator. If you use a calculator, be sure to set the mode correctly. For example, to find

tan(5p>4)1 tan 45°1 cos(5p>4) 12>2

cos 45° 12>2

sin(5p>4) 12>2 sin 45° 12>2 (5p>4)pp>4

5p>4

FIGURE 11

The reference angle for is , or 45°.

p>4

u5p>4

x y

I All positive II

sin (csc)

IV cos (sec) III

tan (cot)

(a) Reference angle is ¨.

(b) Reference angle is π ¨.

(d) Reference angle is 2π ¨.

x y

π Reference angle:4

To evaluate the trigonometric functions in quadrants other than the first quadrant, we use a reference angle. A reference anglefor an angle is the acute angle formed by the -axis and the terminal side of . Reference angles for each quadrant are depicted in Figure 10.

x u

, first set the calculator in radian mode and then enter . The result will be

The number of digits in your answer will depend on the calculator that you use. As we saw in Example 4, the exactvalue of is . Notice that we do not need to use reference angles when we use a calculator.

Graphs of the Trigonometric Functions

Referring once again to the unit circle, which is reproduced in Figure 12, we see that an angle of rad corresponds to one complete revolution on the unit circle. Since is the point where the terminal side of intersects the unit circle, we see that the values of sinuand cosurepeat themselves in subsequent revolutions.

P(x, y)(cosu, sinu) u 2p

12>2 sin(5p>4)

sin5p

4 0.7071068

sin(5p>4) sin(5p>4)

FIGURE 12 The and coordinates of the point

are the same for and u u2p. P y

DEFINITION Periodic Function

A function is periodicif there is a number such that

for all in the domain of . The smallest such number is called the period of .f

p f

f(xp)f(x) p0 f

¨ 2π x y

P(x, y) (cos ¨, sin ¨) 1

The graphs of the six trigonometric functions are shown in Figure 13. Note that we have denoted the independent variable by instead of . Here, the real number denotes the radian measure of an angle. As their graphs indicate, the six trigonometric functions are all periodic. The sine and cosine functions, as well as their reciprocals, the

x x u

Therefore,

and (7a)

and

and (7b)

for every real number and every integer , and we say that the sine and cosine functions are periodic with period .

More generally, we have the following definition of a periodic function.

n u

cos(u2np)cosu sin(u2np)sinu

cos(u2p)cosu sin(u2p)sinu

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FIGURE 13

Graphs of the six trigonometric functions

FIGURE 14

cosecant and secant functions, have period . The period of the tangent and cotangent functions, however, is .

Let’s look more closely at the graphs shown in Figure 13a–b. Notice that the graphs of and oscillate between and . In general, the graphs

of the functions and oscillate between and ,

and we say that their amplitude is . The graphs of and are shown in Figure 14a–b. Observe that the factor 4 in has the effect of

“stretching” the graph of between the values of and 4, whereas the factor in 14 y14sinxhas the effect of “compressing” the graph between 14and .14

4 ysinx

y4 sinx

y14sinx y4 sinx

A y A yA

yAcosx yAsinx

y 1

ycosx ysinx

x y

0 1

π 2

3π 2π

2π 2

3π 2 3π

2 π

2 1

1 1

π x x

y y

0 1

π 2

3π 2π

Domain: x nπ Range: (, ) Period: π Domain: (, )

Range: [1, 1]

Period: 2π Domain: (, )

Range: [1, 1]

Period: 2π

(a) y sin x (b) y cos x (c) y tan x

2 π

2 1

π2 π2 π π π

x y

Domain: x nπ Range: (, 1] 傼 [1, ) Period: 2π

(d) y csc x (e) y sec x (f) y cot x

π 2 π

1 x y

Domain: x nπ Range: (, 1] 傼 [1, ) Period: 2π

π 2

π π

x y

Domain: x nπ Range: (, ) Period: π

π π 2 π

x y

0 1 4

2π π

(a) The graph of y 4 sin x superimposed upon the graph of y sin x

y 4 sin x y sin x

2π π x y

0 1

2π π

(b) The graph of y sin x superimposed upon the graph of y sin x

y sin x y sin x

1 2π π 1

1 4

1 4 1 4

BARTHOLOMEO PITISCUS (1561–1613)

Mathematician Bartholomeo Pitiscus was born in Grunberg, Silesia (now Zielona Gora, Poland), and died on July 2, 1613, in Heidelberg, Germany; beyond this, not much is known of his childhood or of his mathematical education. What is known is that in 1595, when Pitiscus titled his book Trigonometria: sive de solutione triangulo- rum tractatus brevis et perspicuus, he introduced the term trigonometry—a term that would become the recognized name for an entire area of mathematics. His book is divided into three parts, including chapters on plane and spherical geometry, tables for all six of the trigonometric functions, and problems in geodesy, the scien- tific discipline that deals with the measure- ment and representation of the earth.

Historical Biography

Topham/The Image Works

FIGURE 15

We now summarize these definitions.

DEFINITION Period and Amplitude of and The graphs of

and

where and , have A0 B0 period and 2p>B amplitudeA. f(x)AcosBx f(x)AsinBx

AcosBx Asin Bx

EXAMPLE 5 Sketch the graph of .

Solution The function has the form , where and

. This tells us that the amplitude of the graph is 3 and the period is 2p>124p. B12

A3 yAsinBx

y3 sin12 x

Next, let’s compare the graphs of and with the graph of (see Figure 15a–b). Notice here that the factor of 2 has the effect of “speed- ing up” the graph of the cosine: The period is decreased from to . In contrast, the factor of has the effect of “slowing down” the graph of the cosine: The period is increased from to . In general, the period of both and is

if .B0

2p>B 2p 4p ysinBx ycosBx

1 2

2p p ycosx

ycos(x>2) ycos 2x

x y

0 1

(a) The graph of y cos 2x superimposed upon the graph of y cos x

y cos x y cos 2x

2π π 2π

(b) The graph of y cos superimposed upon the graph of y cos x

x 2

x y

0 1

π y cos y cos x

2π π 2π

FIGURE 16

The graph of has

amplitude 3 and period 4p. y3 sin12 x

x y

0 3

Period 4π Amplitude 3

4π2π 2π 4π 6π 8π

Using the graph of the sine curve, we sketch the graph of over one period . (See Figure 16.) Next, the periodic properties of the sine function allow us to extend the graph in either direction by completing another cycle as shown.

[0, 4p]

y3 sin12 x

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The Trigonometric Identities

By comparing the angles and in Figure 17, we see that the points and have the same -coordinates and that their -coordinates differ only in sign. Thus,

(8) and

(9) We conclude that the cosine function is even and the sine function is odd. Similarly, we can show that the cosecant, tangent, and cotangent functions are odd, while the secant function is even. These results are also confirmed by the symmetry of the graph of each function (see Figure 13).

Equations such as Equations (8) and (9) that express a relationship between trigonometric functions are called trigonometric identities.Each identity holds true for every value of in the domain of the specified trigonometric functions.

Referring once again to the point on the unit circle (see Figure 7), we see that the equation can also be written in the form

(10) Note Recall that . In general, is usually written . The same convention applies to the other trigonometric functions.

The additionand subtraction formulasfor the sine and cosine are

(11) and

(12) If we let in Formulas (11) and (12), we obtain the double-angleformulas

(13) and

(14a) (14b) (14c) Solving (14b) and (14c) for and , respectively, we obtain the half-angle formulas

(15) and

(16) These and several other trigonometric identities are summarized in Table 3.

sin2A 1

2 (1cos 2A) cos2A 1

2 (1cos 2A) sin2A

cos2A

12 sin2A 2 cos2A1 cos 2Acos2Asin2A

sin 2A2 sinAcosA AB

cos(AB)cosAcosBsinAsinB sin(AB)sinAcosBcosAsinB

sinnu (sinu)n

sin2u (sinu)2

cos2usin2u1 x2y21

P(x, y) u

sinuy sin(u) cosuxcos(u)

y x

P¿ u P

FIGURE 17

The angles and have the same magnitude but opposite signs.

u u P(x, y)

P(x, y)

x y

Pythagorean identities Half-angle formulas Addition and subtraction formulas

cot2u1csc2u tan2u1sec2u cos2usin2u1

sin2A12(1cos 2A) cos2A12(1cos 2A)

cos(AB)cosAcosBsinAsinB sin(AB)sinAcosBcosAsinB

Double-angle formulas Cofunctions of complementary angles

12 sin2A

cosusin1p2u2

cos 2Acos2Asin2A2 cos2A1

sinucos1p2u2

sin 2A2 sinAcosA TABLE 3 Trigonometric Identities

EXAMPLE 6 Find the solutions of the equation that lie in the interval .

Solution Using the identity (14b), we make the substitution , obtaining

Thus,

and x2p>3,4p>3, 0, and 2pare the solutions in the interval [0, 2p].

cosx 1

2 or cosx1

2 cosx10 or cosx10

(2 cosx1)(cosx1)0 2 cos2xcosx10 (2 cos2x1)cosx0 cos 2xcosx0

cos 2x2 cos2x1 [0, 2p]

cos 2xcosx0

A more complete list of trigometric identities can be found in the reference pages at the back of the book.

Videos for selected exercises are available online atwww.academic.cengage.com/login.

In Exercises 1–8,convert each angle to radian measure.

1. 150° 2. 210° 3. 330° 4. 405°

5. 6. 7. 8.

In Exercises 9–16,convert each angle to degree measure.

9. 10. 11. 12.

13. 14. 15. 16. 11p

13p 3 11p 4

p 6 2

9p 4 5p

6 3p

4 p

495°

75°

225°

120°

In Exercises 17–24,find the exact value of the trigonometric functions at the indicated angle.

17. , , and for

18. , , and for

19. , , and for

20. , , and for

21. , , and for

22. , , and for

23. csct,sect, and for cott t17p>6 a 3p>2 csca

cota cosa

ap csca

tana sina

x5p>6 cscx

cotx sinx

x2p>3 secx

tanx cosx

u p>4 cscu

cosu sinu

up>3 tanu

cosu sinu